Arcavox,
Sorry for the delay. I successfully registered this time. Not sure what was up yesterday, but hopefully it's all good now.
I can see the issue you mention, and it's definitely strange. To Logout, we call a little bit of Javascript code. If there are Javascript errors on your page though, it can prevent some of the Facebook logic from loading, which then prevents us from logging the user out. On that specific page, I'm seeing the following errors (using the Chrome Developer Tools):
user_timeline.json: Resource interpreted as Script but transferred with MIME type application/json.
I don't know what that user_timeline.json file is, but I think it's the cause of the issue. Are you familiar with this, or have any knowledge of it?
Hopefully, this helps you narrow down the cause, but if not, let us know and we'll do what we can to help!