Using language_default() causes magic errors on a multilangual site with different domain names

"And so if you have different domains for the English and Hungarian languages then it would produce different "redirect URL" for every domain - so you should probably have different applications for every domain - meaning different provider settings (application ID/secret, etc.) - is that correct?"

Yes, it's correct. ...at least that's how i did it.

duozersk,

Yes it's needed to implementing hook_variable_info() to publish hybridauth variables on admin/config/regional/i18n/variable admin page ( created by i18n_variable module ( part of i18n module ) ):

/**
 * Implements hook_variable_info().
 */
function MYMODULE_variable_info($options) {
  $variables['hybridauth_provider_Facebook_keys_id'] = array(
    'title' => t('Facebook App ID'),
    'description' => t('Facebook App ID'),
    'localize' => TRUE
  );
  $variables['hybridauth_provider_Facebook_keys_secret'] = array(
    'title' => t('Facebook App Secret'),
    'description' => t('Facebook App Secret'),
    'localize' => TRUE
  );
  $variables['hybridauth_provider_Google_keys_id'] = array(
    'title' => t('Google App ID'),
    'description' => t('Google App ID'),
    'localize' => TRUE
  );
  $variables['hybridauth_provider_Google_keys_secret'] = array(
    'title' => t('Google App Secret'),
    'description' => t('Google App Secret'),
    'localize' => TRUE
  );
  return $variables;
}

It's not the best solution because if i enable a new provider on admin/config/people/hybridauth page than i should extend the hook_variable_info() implementation manually

PS:

The solution above matches if the different domains depend on languages. If the different domains are created by domain access module than it's another case ( i think so )...

PS II.: hmmm... the main question is here the different domain not the different languages, isn't it?