Well, first off: a couple of those links are to old behaviors or bugs that no longer exist (for instance, you cannot log off of just a single site anymore). If you find a bug in this, report it - but first verify that it's actually a bug (read on)...
See, the behavior you're describing isn't related to the use of local storage, since that is already cleared when you log off...
...It's because you're actually logged in on two separate sites. And you're only logging out of one.
If you log in using a Stack Exchange OpenID, you have to log out of SE OpenID as well. (If that link gives you a "page not found" page, then you're already logged out)
If you log in using Google, you have to log out of Google as well.
If you log in using Facebook...
You get the picture.
The fact that most folks don't realize this is one of the big ongoing problems with most major "single signon" systems. Even if we're able to solve this for our own SE OpenID someday, the majority of SE users actually authenticate using Google - which they'll still be logged into even after logging out of Stack Exchange. That's just how it works.