Yes, it's basically the same as "on prem", but instead just a bunch of individual (virtual) servers they are managed by a system that provides typical cloud APIs, such as OpenStack.

It will mean compute resources that can be transparently shared amongst different things (which is where the K8s comes in) as opposed to boxes dedicated to a single service.

The term "private cloud" has existed for easily 15 years, pretty much straight after Google coined the term "cloud computing". The current deifnition of "on-prem" actually changed after cloud computing reached mass adoption. Prior to then, on-prem referred to "in your office" vs. "in a datacenter". These days "on-prem" means "not in public cloud".

on-prem services but running with the tooling that is typically used for cloud computing rather than bare-metal on-prem of old.