I triplechecked for the marketplace thingy and found something:
/var/lib/openhab/marketplace/kar/133947/org.openhab.io.openhabcloud-3.3.0-oh320-SNAPSHOT.kar
That fixed back then some connection issues. I totally forgot about it. After killing it (and some other 3.2.0 corpses) it started to work again for me. Just for somebody having similar issues.
Also if you see some errors like
(HANDLER_MISSING_ERROR): Handler factory not found
It means (at least for me) that the binding for that thing is not installed. For me it went away when I killed the 3.2.0 versions that “patched” some issues in the past. Those made it hopefully to 3.4 now 
Back then I asked the devs, if it would be a problem when the official version will come in the future and they said “no” :D. Apparently it can lead to those issues. I don’t fully get though, why this came with the 3.4.1 → 3.4.4 upgrade. I would consider changes on patch version level not to break something. But not sure if changed deps count into that.
Anyway, thanks for all your support!