diff -u b/core/modules/user/config/install/user.settings.yml b/core/modules/user/config/install/user.settings.yml
--- b/core/modules/user/config/install/user.settings.yml
+++ b/core/modules/user/config/install/user.settings.yml
@@ -1,6 +1,5 @@
 anonymous: Anonymous
 verify_mail: true
-password_register: false
 notify:
   cancel_confirm: true
   password_reset: true
@@ -10,6 +9,7 @@
   register_admin_created: true
   register_no_approval_required: true
   register_pending_approval: true
+  register_password_set: false
 register: visitors
 cancel_method: user_cancel_block
 password_reset_timeout: 86400
diff -u b/core/modules/user/config/schema/user.schema.yml b/core/modules/user/config/schema/user.schema.yml
--- b/core/modules/user/config/schema/user.schema.yml
+++ b/core/modules/user/config/schema/user.schema.yml
@@ -10,7 +10,7 @@
     verify_mail:
       type: boolean
       label: 'Require email verification when a visitor creates an account'
-    password_register:
+    register_password_set:
       type: boolean
       label: 'Require people to choose a password during registration.'
     notify:
diff -u b/core/modules/user/src/AccountForm.php b/core/modules/user/src/AccountForm.php
--- b/core/modules/user/src/AccountForm.php
+++ b/core/modules/user/src/AccountForm.php
@@ -162,7 +162,7 @@
         }
       }
     }
-    elseif (!$config->get('verify_mail') || $config->get('password_register') || $admin_create) {
+    elseif (!$config->get('verify_mail') || $config->get('register_password_set') || $admin_create) {
       $form['account']['pass'] = [
         '#type' => 'password_confirm',
         '#size' => 25,
diff -u b/core/modules/user/src/AccountSettingsForm.php b/core/modules/user/src/AccountSettingsForm.php
--- b/core/modules/user/src/AccountSettingsForm.php
+++ b/core/modules/user/src/AccountSettingsForm.php
@@ -163,11 +163,11 @@
       '#default_value' => $config->get('verify_mail'),
       '#description' => $this->t('New users will be required to validate their email address prior to logging into the site, and will be assigned a system-generated password. With this setting disabled, users will be logged in immediately upon registering, and may select their own passwords during registration.'),
     ];
-    $form['registration_cancellation']['user_password_register'] = [
+    $form['registration_cancellation']['register_password_set'] = [
       '#type' => 'checkbox',
       '#title' => $this->t('Require people to choose a password during registration.'),
       '#description' => $this->t('If <em>Require e-mail verification</em> is disabled, this setting is automatically enabled.'),
-      '#default_value' => $config->get('password_register'),
+      '#default_value' => $config->get('register_password_set'),
       '#states' => [
         // Disable this option if email_verification is unchecked.
         'disabled' => [
@@ -492,8 +492,8 @@
       ->set('register', $form_state->getValue('user_register'))
       ->set('password_strength', $form_state->getValue('user_password_strength'))
       ->set('verify_mail', $form_state->getValue('user_email_verification'))
-      ->set('password_register', $form_state->getValue('user_password_register'))
       ->set('cancel_method', $form_state->getValue('user_cancel_method'))
+      ->set('register_password_set', $form_state->getValue('register_password_set'))
       ->set('notify.status_activated', $form_state->getValue('user_mail_status_activated_notify'))
       ->set('notify.status_blocked', $form_state->getValue('user_mail_status_blocked_notify'))
       ->set('notify.status_canceled', $form_state->getValue('user_mail_status_canceled_notify'))
diff -u b/core/modules/user/src/Entity/User.php b/core/modules/user/src/Entity/User.php
--- b/core/modules/user/src/Entity/User.php
+++ b/core/modules/user/src/Entity/User.php
@@ -144,7 +144,7 @@
         // if the account never signed in and if we are
         // called from the user_register_form and password_register
         // is set to enabled.
-        if (!$this->login->value && \Drupal::config('user.settings')->get('password_register') && !$this->original->status->value && $this->status->value) {
+        if (!$this->login->value && \Drupal::config('user.settings')->get('register_password_set') && !$this->original->status->value && $this->status->value) {
           $op = 'register_password_set_activation';
         }
 
diff -u b/core/modules/user/src/RegisterForm.php b/core/modules/user/src/RegisterForm.php
--- b/core/modules/user/src/RegisterForm.php
+++ b/core/modules/user/src/RegisterForm.php
@@ -71,7 +71,7 @@
     $config = $this->config('user.settings');
     $admin = $form_state->getValue('administer_users');
 
-    if (!$config->get('verify_mail') || ($config->get('verify_mail') && $config->get('password_register')) || $admin) {
+    if (!$config->get('verify_mail') || ($config->get('verify_mail') && $config->get('register_password_set')) || $admin) {
       $pass = $form_state->getValue('pass');
     }
     else {
@@ -81,7 +81,7 @@
     // If we are not an admin and we try to register and password_register
     // is set, make sure the status is set to disabled before we save
     // the newly created account.
-    if ($config->get('verify_mail') && $config->get('password_register') && !$form_state->getValue('uid') && !$admin) {
+    if ($config->get('verify_mail') && $config->get('register_password_set') && !$form_state->getValue('uid') && !$admin) {
       $form_state->setValue('status', 0);
     }
 
@@ -128,9 +128,9 @@
       ]));
     }
     // Email verification enabled, but users set a password during registration.
-    elseif (!$admin && $config->get('register') == UserInterface::REGISTER_VISITORS && $config->get('password_register') && !$account->isActive()) {
+    elseif (!$admin && $config->get('register') == UserInterface::REGISTER_VISITORS && $config->get('register_password_set') && !$account->isActive()) {
       // Notify the user.
-      _user_mail_notify('register_password_set', $account);
+      _user_mail_notify('register_pending_approval', $account);
       $this->messenger()->addStatus($this->t('A welcome message with further instructions has been sent to your email address.'));
       $form_state->setRedirect('<front>');
     }
diff -u b/core/modules/user/tests/src/Functional/UserRegistrationTest.php b/core/modules/user/tests/src/Functional/UserRegistrationTest.php
--- b/core/modules/user/tests/src/Functional/UserRegistrationTest.php
+++ b/core/modules/user/tests/src/Functional/UserRegistrationTest.php
@@ -21,9 +21,7 @@
 
   use StringTranslationTrait;
 
-  use AssertMailTrait {
-    getMails as drupalGetMails;
-  }
+  use AssertMailTrait;
 
   /**
    * Modules to enable.
@@ -59,7 +57,7 @@
     $edit['name'] = $name = $this->randomMachineName();
     $edit['mail'] = $mail = $edit['name'] . '@example.com';
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('A welcome message with further instructions has been sent to your email address.'), 'User registered successfully.');
+    $this->assertSession()->responseContains($this->t('A welcome message with further instructions has been sent to your email address.'));
 
     /** @var EntityStorageInterface $storage */
     $storage = $this->container->get('entity_type.manager')->getStorage('user');
@@ -106,7 +104,7 @@
     $edit['pass[pass1]'] = '99999.0';
     $edit['pass[pass2]'] = '99999';
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('The specified passwords do not match.'), 'Typing mismatched passwords displays an error message.');
+    $this->assertSession()->responseContains($this->t('The specified passwords do not match.'));
 
     // Enter a correct password.
     $edit['pass[pass1]'] = $new_pass = $this->randomMachineName();
@@ -119,7 +117,7 @@
       ->loadByProperties(['name' => $name, 'mail' => $mail]);
     $new_user = reset($accounts);
     $this->assertNotNull($new_user, 'New account successfully created with matching passwords.');
-    $this->assertSession()->responseContains($this->t('Registration successful. You are now logged in.'), 'Users are logged in after registering.');
+    $this->assertSession()->responseContains($this->t('Registration successful. You are now logged in.'));
     $this->drupalLogout();
 
     // Allow registration by site visitors, but require administrator approval.
@@ -132,7 +130,7 @@
     $edit['pass[pass1]'] = $pass = $this->randomMachineName();
     $edit['pass[pass2]'] = $pass;
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('Thank you for applying for an account. Your account is currently pending approval by the site administrator.'), 'Users are notified of pending approval');
+    $this->assertSession()->responseContains($this->t('Thank you for applying for an account. Your account is currently pending approval by the site administrator.'));
 
     // Try to log in before administrator approval.
     $this->drupalGet('user/login');
@@ -142,7 +140,7 @@
       'pass' => $pass,
     ];
     $this->submitForm($auth, $this->t('Log in'));
-    $this->assertSession()->responseContains($this->t('The username @name has not been activated or is blocked.', ['@name' => $name]), 'User cannot log in yet.');
+    $this->assertSession()->pageTextContains(t('The username @name has not been activated or is blocked.', ['@name' => $name]));
 
     // Activate the new account.
     $accounts = $this->container->get('entity_type.manager')->getStorage('user')
@@ -162,7 +160,7 @@
     $this->drupalGet('user/login');
     $this->assertSession()->statusCodeEquals(200);
     $this->submitForm($auth, $this->t('Log in'));
-    $this->assertSession()->responseContains($this->t('Member for'), 'User can log in after administrator approval.');
+    $this->assertSession()->responseContains($this->t('Member for'));
   }
 
   /**
@@ -175,7 +173,8 @@
     $this->config('user.settings')
       ->set('verify_mail', TRUE)
       ->set('register', UserInterface::REGISTER_VISITORS)
-      ->set('password_register', TRUE)
+      ->set('register_password_set', TRUE)
+      ->set('notify.register_pending_approval', TRUE)
       ->save();
 
     $edit = [];
@@ -188,7 +187,7 @@
     $this->drupalGet('user/register');
     $this->assertSession()->statusCodeEquals(200);
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('A welcome message with further instructions has been sent to your email address.'), 'Send e-mail to user after registering.');
+    $this->assertSession()->responseContains($this->t('A welcome message with further instructions has been sent to your email address.'));
 
     // Make sure the user is still blocked.
     $this->container->get('entity_type.manager')->getStorage('user')->resetCache();
@@ -206,20 +205,25 @@
     $this->drupalGet('user/login');
     $this->assertSession()->statusCodeEquals(200);
     $this->submitForm($edit2, $this->t('Log in'));
-    $this->assertSession()->responseContains($this->t('The username %name has not been activated or is blocked.', ['%name' => $name]), $this->t('User cannot login yet.'));
+    $this->assertSession()->responseContains($this->t('The username %name has not been activated or is blocked.', ['%name' => $name]));
 
     // Try to activate the user.
-    $_emails = $this->drupalGetMails();
+    $new_user->activate();
+    $new_user->save();
+    $_emails = $this->getMails();
     $this->assertCount(2, $_emails);
-    $email = end($_emails);
+    $email = reset($_emails);
     $urls = [];
     preg_match('#.+user/reset/.+#', $email['body'], $urls);
+    $this->drupalGet($urls[0]);
+    $this->drupalPostForm(NULL, [], t('Log in'));
 
-    return $urls[0];
+    // Change the password.
+    $password = \Drupal::service('password_generator')->generate();
+    $edit = ['pass[pass1]' => $password, 'pass[pass2]' => $password];
+    $this->drupalPostForm(NULL, $edit, t('Save'));
+    $this->assertText(t('The changes have been saved.'), 'Password changed.');
 
-    $_uid = $new_user->id();
-    $this->drupalGet($urls[0]);
-    $this->assertSession()->responseContains($this->t('You have just used your one-time login link. Your account is now active.'), $this->t('User account activated.'));
   }
 
   /**
@@ -244,7 +248,7 @@
     $this->drupalGet('user/register');
     $this->assertSession()->statusCodeEquals(200);
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('The email address @email is already taken.', ['@email' => $duplicate_user->getEmail()]), 'Supplying an exact duplicate email address displays an error message');
+    $this->assertSession()->pageTextContains($this->t('The email address @email is already taken.', ['@email' => $duplicate_user->getEmail()]));
 
     // Attempt to bypass duplicate email validation by adding spaces.
     $edit['mail'] = '   ' . $duplicate_user->getEmail() . '   ';
@@ -252,7 +256,7 @@
     $this->drupalGet('user/register');
     $this->assertSession()->statusCodeEquals(200);
     $this->submitForm($edit, $this->t('Create new account'));
-    $this->assertSession()->responseContains($this->t('The email address @email is already taken.', ['@email' => $duplicate_user->getEmail()]), 'Supplying a duplicate email address with added whitespace displays an error message');
+    $this->assertSession()->pageTextContains($this->t('The email address @email is already taken.', ['@email' => $duplicate_user->getEmail()]));
   }
 
   /**
@@ -426,7 +430,7 @@
 
     // Check that the field does not appear on the registration form.
     $this->drupalGet('user/register');
-    $this->assertSession()->responseNotContains($field->label(), 'The field does not appear on user registration form');
+    $this->assertSession()->responseNotContains($field->label());
     $this->assertSession()->responseHeaderContains('X-Drupal-Cache-Tags', 'config:core.entity_form_display.user.user.register');
     $this->assertSession()->responseHeaderContains('X-Drupal-Cache-Tags', 'config:user.settings');
 
@@ -436,7 +440,7 @@
       ->save();
 
     $this->drupalGet('user/register');
-    $this->assertSession()->responseContains($field->label(), 'The field appears on user registration form');
+    $this->assertSession()->responseContains($field->label());
     $this->assertRegistrationFormCacheTagsWithUserFields();
 
     // Check that validation errors are correctly reported.
diff -u b/core/modules/user/user.module b/core/modules/user/user.module
--- b/core/modules/user/user.module
+++ b/core/modules/user/user.module
@@ -1020,7 +1020,6 @@
  *     self-registers.
  *   - 'register_pending_approval': Welcome message, user pending admin
  *     approval.
- *   - 'register_password_set': Welcome message, password set.
  *   - 'register_password_set_activation': Activation message, password set.
  *   - 'password_reset': Password recovery request.
  *   - 'status_activated': Account activated.