My guess is that you have the "Auto Login Facebook Users" set to "Yes". In this case, since you're using a different Logout button that isn't logging the user out of Facebook, when the page reloads, the user is being automatically logged back in. Try changing that setting if you aren't using the JFBCLogin module, and you should be all fixed up.
On another note, when logging in, there should be a system message that notifies the user they were automatically logged in. You may want to check your template to ensure that it's displaying the Joomla system messages.