screen log in log out

Not sure if this is what you’re looking for but have you tried the Peter’s Login Redirect plugin? It allows you to redirect users to any page on login and a different page on logout. I’m not sure how well it plays with multisite either.

https://www.ads-software.com/plugins/peters-login-redirect/

Are you trying to just style the WP logout page, want to change the logout page altogether, or trying to style the plugin’s logout page?