Hah, finally found the solution for that curious problem.
After 3.5 days of deep code inspection and debugging, i did one last check. I inspected the two databases on tables related to access, policies, users, groups and the like.
Also on tables related to session data. There i came across the table modx_session.
On the windows system this table had entries, on the debian install this table was empty - hmm, why on earth ..., what the heck is ...
Something with sessions seems to be different between the two MODx installs. So i checked the complete session based environment within MODx itself and the underlying PHP environment.
The session parameters under System Settings were equal, but - two session parameters within the PHP environment were different. The first parameter was session.save_path - this can be ignored, the second parameter was session.auto_start - on the windows install it was Off, on the debian install it was On. Because i had no access to the php.ini on the debian system (its a shared host), i decided to put this parameter into a .user.ini file (PHP is implemented as a FastCGI-Process) - and tada, it worked.
Now all access rights are as they should be - as i wanted them to be.
It would be great, if this fact would find its way into the MODx manuals. I mean it seems to be essential for MODx to have this session value being set to Off in order to work properly when it comes to the permission system and acl. And by the way it would be nice, if someone can clarify why this setting completely disables/bypasses the overall pemission system of MODx.