Strange. It's still working fine for me. Said I wasn't logged in, so I logged in and checked and work, and I logged out again and checked and it worked. I wonder if you're using a proxy that is interfering somehow? I'm assuming it's not an addon as you said it's the same in both Chrome and Firefox?
I'm sure this isn't what you're thinking, but just to double check... You don't think that you're logged out of twitter just because it's not open anymore do you? If you log in, and then close the tab without logging out, then you're still logged in...
I know I wasn't logged in because I don't even have a twitter account :). But, your assumption that this may be a proxy issue is almost certainly right, since I accessed the page from my work computer. I tried it now from my home computer and everything checks out - it doesn't show that I'm logged in anywhere except where I actually am.
For the Twitter test, the HTTP response code is an error code if you're logged in. So if your work place blocks Twitter and returns an error code like 403 or something, then you will appear to be logged in.
The test could easily be modified so it checks some other url first to make sure twitter isn't generally blocked.
The intention of the article was to describe a general technique, rather than to provide some complete fully functional tests. Although they do work for the vast majority of people.
I'm sure this isn't what you're thinking, but just to double check... You don't think that you're logged out of twitter just because it's not open anymore do you? If you log in, and then close the tab without logging out, then you're still logged in...