I think a lot of people would consider having a rack at a colo to be "on-prem". I usually thought of that term meaning you're renting space and the services provided are redundant/reliable power, a public internet drop (or multiple), and physical security. Where you need to rack it yourself, run your own networking gear, etc.

I don't see how that's too functionally different from having a suite at a commercial office building, save for the fact the power and data isn't going to be nearly as reliable. It's just your "suite" is only a bit more than 19" wide.