From watching the linked screen recording, my immediate guess as to what might be happening here is:

• Tapping the ‘log out’ link appears to result in the user being logged-out in the background (i.e., while they’re on the same page).
• After having logged the user out in the background, MediaWiki then appears to try to redirect the user to a ‘you have been logged out’ page.
• Because the user has made changes in the editor, another part of MediaWiki might then trigger the browser warning of ‘do you want to leave this page?’.
• Tapping ‘stay’ might prevent the user from being redirected to a ‘you have been logged out’ page. However, because (at this point) the user may have already been logged-out in the background, it might not prevent the logging-out itself from taking place.

I should note that I haven’t looked at the source code that might be causing this behaviour - this is my guess purely based on having watched the screen recording, so it definitely isn't guaranteed to be correct!