The "log out" functionality of Stack Exchange appears to be broken. You can reproduce this with the following steps:
Logging in:
- Click "log in" on the top bar
- Click "log in with Stack Exchange"
- Enter your username and password, then click "Sign in"
Logging out:1
- Click the drop down menu at the top, then click "log out"
- Check "log out on all devices" and click "Log Out"
Logging in again:
- Click "log in" on the top bar
- Click "log in with Stack Exchange"
When logging in the second time, you will not be prompted for a username or password. This means that you were never actually logged out. You can even close and reopen the browser in-between steps 5 and 6 and get the same result.
Obviously this is a security concern for users who log in on public machines.
The only solution I am aware of at the moment is to clear the cookies after logging out.
This question has been asked before ([1] [2]). It seems that the usual recommendation is to log out of your OpenID provider — but in my case, I'm logging in with Stack Exchange (rather than Facebook, Google, etc), and I can't find another log out link.
1 These are the same instructions detailed here