Private cloud usually refers to that same abstraction layer. Hardware provisioning is abstracted away from use and available on-demand from a pool, and likewise deployment works automatically via the same or an equivalent API.
That's very different from the operation of a typical company-owned datacenter from a couple of decades ago.
cloud - someone else's computers
private cloud - someone else's computers that you own???? or that nobody else uses, but in this context you also own???