I am using Safari on macOS 11.3 (Big Sur).

* I click Logout.
* The address bar switches to http://www.rchess.de/forum/user/logout.page
* There appears a progress bar.
* After a few moments the address bar switches back to the page I was on.
* The progress bar remains but does not move.
* It never logs me out.

Maybe instead you can set an expiration/timeout on the login?

P.S. Your BB Code is not behaving like other message boards I have used.