Okay. Here is a thread from two summers ago about this very matter.
That thread has some theories and possible connections, but I didn't see any explanation or resolution for this matter in that thread. And regarding jeremy's comment in post #6 that what I'm seeing is not real, that I really am logged out when I see this stuff, well that's wrong. I just did the whole thing again and came right back here to post this without having to log in.
P.S.: One clarification of what I'm seeing, just for accuracy... Click [Log Out]. Click the OK button that I'm sure. The "Are you sure" dialog closes but I'm still shown logged in. Click [Log Out] again. Click the OK button again. Now
is when it looks like I am logged out for one second followed by the black banner for a couple of more seconds. Then the black banner vanishes and I'm shown logged in with "Welcome, stoat. [Log Out]" back again. And it is at this point that every click on [Log Out] produces the one-second-logout-black-banner-log-in thing, and it can be repeated infinitely. And I really am logged in. I posted this after what I just described. If I close the Firefox browser, restart it, and return, then I have to log in again.