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Thanks, Alexander, for the suggested answer. It didn't work for me. Still the "The server your are trying..." message.
Also also tried to combine it with Alex Melching's solution. Strange things happen. Word crashes after logging in with the school account, then restarts, all seems fine but it reports a server error from trying to log in to SharePoint (actually ODFB).
No problem to login to SharePoint via the browser.
Then all returns to normal, after restarting Word once again, but still no sync. Btw, this is a fresh Office 2016 install. Behavior is the same to the Office 2016 upgrade on the other laptop.
It looks like some particularities apply. For instance, I still get German language system messages when logging in to Microsoft accounts. I also use the same e-Mail for both the Microsoft and the Work account (different passwords). It looks like some errant undefined parameter somewhere sometimes switching to one then to another random value in Office 2016.