In brief: The logout page should tell users that, in order to make the logout "stick", they also need to log out of their OpenID provider. It should tell this very clearly and explicitly, and preferably several times. Also, global logout would be great.
This is actually a bunch of closely related feature requests rolled into one post, because I feel they're close enough in scope and purpose that splitting them into several requests would be just silly. I'd be happy to see any subset of them implemented, but the more, the better, IMO.
1. Add a note about OpenID to the logout page
The current logout page is extremely spartan: all it has in one button and a very short paragraph of text explaining what it does.
I suggest that this paragraph should be clarified to make it clear that it only applies to one specific site, and amended to tell users that they also need to log out of their OpenID provider, something like this (additions underlined):
(It might be better if the link to "your OpenID provider" actually used the name of the provider, at least if it's known. We could at least do this for SE OpenID and for major providers like Google and Facebook.)
I believe this feature is important even if the other features below are implemented too. Why? Because if the user is leaving a public computer in a hurry (maybe they're on a shared computer at school, and class is just starting), they're likely to click the "Log Out" button, assume that it logs them out completely and just run off without carefully observing the results.
Thus, if the "Log Out" button doesn't do what they'd naïvely expect, we need to warn the users before they click it.
Edit: Part 1 is now mostly status-completed, except for the link. Thanks, Laura!
2. After logout, show a page telling users what they need to do next
Currently, after logging out, the user is just redirected to the front page of the site. (To add some extra confusion, due to browser caching, this page may sometimes show them as still logged in.)
I suggest that the user should instead be shown a page explicitly confirming that they've logged out, and telling them if they're still logged into their OpenID provider and/or any other SE sites, e.g. like this:
(Verifying the OpenID provider login status should be doable using an immediate OpenID request.)
Also, obviously, it should be verified that the claims on this page (particularly the "You will not be logged in..." part) are actually true.
3. Streamline the logout process for SE OpenID specifically
To an average user, who's not familiar with the differences between "Stack Overflow" and "Stack Exchange" and "Stack Exchange OpenID", the second paragraph in the mockup above would look really confusing. (It works OK if you replace "Stack Exchange OpenID" with "Google" or "Facebook", though, since nobody expects those sites to be the same as SO.)
Instead of telling users to go to http://openid.stackexchange.com and log out of there, we should just give them a "Log out of SE OpenID" button on the per-site logout page. In general, OpenID does not provide a way for a Relying Party to embed such a button, but since in this case SE hosts both the OpenID provider and the RP, surely such integration could be arranged.
4. Provide a button to log out of all Stack Exchange sites with one click
Obviously, expecting a user in a hurry to log out of a dozen SE sites one at a time is ridiculous. There should be a single button to log the user out of all of them at once.
Actually, that's already supposed to exist. The only problem is that it doesn't seem to work. (I just clicked that button. I'm still logged in.) Please make it work, and include it on the logout page.
In fact, arguably, this should be the default: the "log out" link in the top bar should take the user to a page that will let them log out with one click from all SE sites, and, if they're using SE OpenID, from that as well. Per-site logout could still be offered as an option, e.g. hidden behind a second link, but it's not what most users want, and thus it should not be presented as the default.
The resulting logout form could look something like this (for SE OpenID users):
or like this (for users of other OpenID providers):