There really isn’t a logout feature, because once logged in, the server knows your device by its wifi mac address. The “login” feature simply associates that mac address with your FlightAware username/account.
Subsequent connections do not re-send your username and password. Instead, it just sends your mac address only, and the server responds with a message to the device indicating that it knows who you are, and simply disables the login button. The app does not store any of your FlightAware account credentials in any way. The app has no access to your FlightAware account beyond linking your device’s mac address with your account.
In the next version of the app, the text will change to “Logged in,” but there still isn’t a “logout”.
Does that make sense?