[Zircom]

>I think people believe that "on prem" means actually racking the servers in your closet, but you can get datacenter space with fantastic power, cooling, and security almost anywhere these days.

That's because that is what on prem means. What you're describing is colocating.

[pclmulqdq]

When clouds define "on-prem" in opposition to their services (for sales purposes), colo facilities are lumped into that bucket. They're not exactly wrong, except a rack at a colo is an extension of your premises with a landlord who understands your needs.