If you've closed the browser (or cleared the cookies or hit the logout button) the session cookies are gone and you aren't really logged in any more, just LJ has no way of knowing that yet. If someone opens a browser they won't be logged in as you. Of course if you've ticked the box for LJ to remember your login, you will still be logged in and will need to hit the logout button when you're done.

You can use the sessions page I linked to remotely log out things you know you accidentally left open or browsers where you no longer want your login details stored.