Peter and John, this difference most likely has to do with checking the "remember my login on this computer" checkbox when you log in to the site. If you keep that checked, closing your browser, shutting down your computer, getting disconnected, etc., will not log you out of the site. That's the same as any other site that has that option. So John, you probably are not checking that checkbox, while Peter is.
While you should not check that checkbox on a shared computer, such as at a library, there's nothing particularly insecure about checking it at home.