Why is there no direct way to sign out of the Apple store web site?

I logged into the Apple store web site http://store.apple.com/us/account/home to change my password. I could not find a link to do that, so I wanted to log out. I could not find any way to log out. I contacted Apple Chat. The Chat person said the following: "To log out of the Apple Online Store, you are going to need to close all of the pages that are associated with the Apple Online Store. After that, you are going to need to clear your cache and cookies of your Internet browser."

This would seem like a security problem, for example, if I were to login at a hotel computer to check something about my account such as a purchase. If I am not a computer expert and can't find an obvious way to log out, then I would be completely exposing my Apple account to the next user to sit down at that computer. Being logged in, they could do anything to my account they want.

I would appreciate comments on this. I must be overlooking something or maybe did not communicate effectively with the Apple Chat person.