How do you actually log out of Messenger? I genuinely cannot find the button

It is annoyingly buried. Tap your profile photo top left, scroll all the way to the very bottom, and there is a Log out row under the account section. On some versions you have to go through the account switcher and remove the account from there instead.

This is exactly why I read the forum before touching settings, thanks all.

One thing to add: update the app first, half of these issues are just old builds.

If you truly have no log out option (older builds hide it), the reliable way is Account Center, switch to the account, then Log out, or just remove the account from the switcher. Clearing the app data also signs you out as a last resort.