Is it possible to add a logout button or script to the server-manager and the user-manager?
It's not possible, without changing the authentication scheme used by the manager. HTTP basic authentication doesn't allow the server to tell the browser to forget the credentials it is using. The only way for the browser to lose them is for the browser to be shut down.
To have what you'd want the manager would need to have session based authentication, and that's a non-trivial change.