Domoticz doesn't matter here at this point.
The logs I posted are from Arduino IDE Serial Monitor.
However, this can be closed without me yet again understanding what happened.
I made a mistake of changing two things at the same time.
I disconnected everything and left it for an hour or so.
I reconnected and launched MYSController - and the damned thing just started working at one. I didn't have to change any settings, include, click anything. It just worked.
Now, Domoticz works too, of course. All that's left is to start soldering on a proper board instead of the bread board I was using.
Thanks for trying @AWI