Chances are, other forum sites you frequent are written in PHP and the software handles logging-in and logging-out directly, itself.
Those are apples. RMN is an orange.
Our forum software, which we brought into use in 2002 or possibly even as early as 1998, does not do logging-in - that's why the /members area is secured by way of Apache's Basic Authentication method.
That method works the way it works. To get it to log you out, I had to use a little magic to convince the server that you're trying to log in with an invalid username or password. The server, just as it should, sends a rejection code. The browser, just as it should, then _forgets_ the correct terms you'd entered earlier, which is what we want, and, just as it should, brings up its dialog box, inviting you to enter a different username and password combination to send to the server.
Is that confusing? Yes, a little. But most any CGI Reporter or Reading Room Agent would soon figure it out, and life would go on.
The posting name is remembered after using the "Log Out" link because it's stored in the same cookie that lets the software know which posts are 'NEW' to you. Confusing? Yes, a little. Yet, again, most would figure it out soon enough, and life would go on. I will, however, look into it for you and see if I can 'fix' it to your satisfaction (chiefly because it's a puzzle and I'm a Virgo. )