Module does not work correctly for node type "book"

I have additional information:
The module of course is enabled, but my statement is ONLY true for nodes, which belong to a book as it seems.
Here is what I did:
I create a category with two terms "a" and "b". As node types for the category I selected only book nodes. Now, I created a new book page, with content and select the first term of the category I created ("a"). I allow for everyone to read it. Everything is fine here.
I add another book page, also assign category "a", fill it with some content. As parent of this page I select the page I already created. This page is NOT visible by anonymous users. Sadly, when I log out, its content is still visible. You can see it here: http://secrets.pirotesse.de/node/4
Yet - I also created a normal page: http://secrets.pirotesse.de/node/5 - this page does not belong to any category and is not a book page. It's only a page. The permissions work correctly here.

Set priority to normal due to only book pages affected.

Okay - here's some more information:
I toyed around with phpmyadmin and the node privacy byrole module some more and found something out:
The first thing I tried is create a normal page with content, and not allowing anonymous users to see it. I know this works.
verified it, by logging out and trying to access the page -> access denied. Wonderful.
The next step was now: Assign this page to a book by editing it again and using the outline feature. And see: It still works! Anonymous users cannot acces the page. Now, what's the difference?
Checking the "node" table with mysql I saw a difference: The type is different. For nodes created as book pages the type is "book". For nodes created as normal pages the type is "page". I toyed around by changing the value of "type" for a node entry from "book" to "page" and see: anonymous users can't access it anymore. Maybe this information helps?

Here's a workaround:

To add nodes to a book, you can "simply" add a new node as page - and not as book page! Fill it with content and submit it. After you have done that, you can go back to that page, edit it, and add it to the book via the outline function.

Just re-setting the title, so bug description matches content. Workaround see above.
Maybe this is a problem in drupal or the book module rather than in this module? I don't know enough about PHP and drupal to understand the way things work.