Doesn't quite role off the tongue though, not even a good acronym. Maybe Personal Data Centre (pdc)?
There's some cross over with r/selfhosted but that generally includes people hosting things on VPS/Bare metal.
On-premises means in a physical location, i.e. data center (or garage) in this context.
On premise would mean based on an argument/inference.
Just today I saw a post (on Reddit) 'Mauling' over whether to get a Framework 16 or not (I assume that English was not their native tongue). Me? I would have just let the bear have it ;-p