The fix for this sort of thing is pretty simple. Just keep a nonce on a user (the "user version" if you will) that increments every time a user makes a change to their login credentials, store that nonce in their cookie, and if the one in the cookie is lower than the one on the server treat it as a logout.

That said, this is sort of an unserious security "issue" for most people between not really caring about the security of their account and reasonably short cookie lifetime settings.