Playwright: Make it easier to test users with logged out state

[Mamaduka]

Based on the conversation in https://github.com/WordPress/gutenberg/pull/58171#discussion_r1464748394. Somewhat related issue - #55751.

Currently, it isn't straightforward to perform user logout action in e2e tests without affecting other tests in the worker. Let's make it easier to log in and log out of various users during the tests in a contained manner.

cc @WunderBart, @kevin940726