That helped a little, maybe.
I did both:
With regards to (2), I could not find key PlexOnlineHome, but I removed the other three entries.
There was no difference in behavior though when accessing my PMS instance via the hostname; it would require me to login first and then my media was not listed as before.
If however I used the IP address instead to connect to my PMS instance, I would see my media right away, but then under PMS > Settings > General the warning This server is unclaimed and not secure shows up and when I click on CLAIM SERVER I’m getting error
There was a problem signing in to Plex.
If I cannot claim my own PMS then I’m stuck with just a different problem.
Also, how can it be that I get different behavior if using http://<PMS-IP>:32400/web/ vs http://<PMS-hostname>:32400/web/ if the <PMS-hostname> is mapped to <PMS-IP> in all my /etc/hosts?