Unlikely, auth mediation never gives out your password as part of logging into some service, it gets asks to confirm you are logged in and if so, to issue a tem

Unlikely, auth mediation never gives out your password as part of logging into some service, it gets asks to confirm you are logged in and if so, to issue a temporary auth tokens that acts as proof that Google thinks your session is authenticated. So if you explicitly log out, then that token should get invalidated, and logging back in should yield a new auth token.