@glyph … and woe betide you if you have the misfortune to both (a) be a teacher in a school system that uses MS infrastructure, and (b) have children studying in the same school system. This appears to be a use case that MS authentication is unable to account for. It doesn’t matter what you’re trying to do - you’re logged into the “other” system, and trying to correct things only makes things worse. Incognito browsing and/or completely separate browsers appears to be the only solution.
Ask me how I know.