I would define hosting in the cloud as "self-hosting" because you perform the act of hosting that thing and thus have full control over it.

Yes it's not the same as hosting as your own hardware but I would argue that it accomplishes the same goals. The goal is to get other people out of your data. With a "service" the underlying company has every incentive to look at your data and use that to improve it's service, but when you host something yourself with a provider, it's in that providers best interest to never look at your data.

For me it all comes down to the goals of the people your are doing business with. With Cloudflare at least, their goal is to just upsell me on hosting equipment, they couldn't give a rats ass what I'm hosting on that hardware so long it's not illegal.

Can “on-prem” solve this debate?

You can self-host on-prem or in the cloud.