I just dealt with this situation and found this thread. Considering myself a tech savvy person, I can't believe I needed to search for "how to log out of my Apple store account"...which resulted in pretty much finding out that "you can't".
My wife and I were trying to check the status of orders with our separate accounts and it boggles my mind that there is seemingly no way to manually sign/log out. I first logged into the "Track an Order" page and now every time we go back to that page it automatically shows my account info.
And no, the answer to this should never be "wait X minutes" or "close and reopen browser". This is
Apple we're talking about; the masters of intuitive UI that even tech dummies can use, right?