Basically what you just outlined, is Facebook logging in, once you've already logged in once. A decently designed website would know not to ask for authorization again but just retrieve the correct data based on your previous login. Technically, you only have to make sure you're logged in to Facebook once on each of your devices. Once you've created an account and authorized it to use your Facebook credentials, you don't need to repeat that process. Next time you log in you just click "log in" and the work is done for you :P