placeholder - working on refining this answer to meet SE standards… (I began by posting links that I thought had the essential info … but actually didn’t…)
Update: WOAH! This is way more complicated than I thought! Even after finding and reading the info at the links below.
Facebook is deceptive. Even after ending ALL Messenger app sessions, I could still log back in in the app without my password and without 2 factor authentication (which I DO have enabled!) I would try to use the app and it would let me right in and a new session would appear. And after killing that session, the above would happen again. I tried killing ALL sessions, and that did the trick. At least this once.
I found info at two facebook.com web pages that I thought provided the answer working on making it a proper answer.
Important: Note that at each of these pages, there is a pull-down that customizes the information for the UI in use (User Interface, I.e. where you are logged in) - Android, iPhone or iPad app, or browser (Basic Mobile, Mobile or Computer). Be sure to choose carefully and note the differences for the UI or UIs you are using.
The two web pages are:
The main one: https://mobile.facebook.com/help/211990645501187/
(and the second, which links to the main one.
There is no fb page with the answer to this question for the Messenger apps.
These DO (sort of) answer, respectively :
- How do I log out of Facebook on another computer, phone or tablet? and (astonishingly)
- Can I log out of Messenger?
First, they discourage you by saying, misleadingly, “You can't log out of an account on Messenger” ! Then way, way below, after indroducing 5 distracting, lame suggestions, they say “you can always end an active session from your Facebook app's Security Settings”. How? It seems they make it different in each UI, so use the URL above - if it still works, and choose the UI that matches.
You can manage where you’re logged in from your Security and Login Settings. The WHERE YOU'RE LOGGED IN section lists where you’re currently logged in. Each entry includes a date, time, location and device type.
To end a session:
- Hunt for, or use the URL above to find your way to, Security and Login.
- Go to the section WHERE YOU'RE LOGGED IN.
- YMMV. You will likely ned to ignore their misleading/false claim that “Tapping Log Out will immediately log you off of Facebook on that computer, phone or tablet." Kill all sessions.