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.

To me, "self-hosted" means, either I use a rented hardware or I use a purchased hardware.

If the software can be switched to a rented or purchased hardware (I do not want Cloudflare's offering) then it is "self-host-able" IMO.

I think there are open source Cloudflare alternative hosting software.

I agree.

I would describe it as externally hosted yet managed specifically because someone else is most likely, whether they admit to it or not, recording some sort of logs on my service and data and that is a choice that I do not own or, in some cases, have access to.

It is confusing, I agree. IMO we should have "self-running" and "self-hosting". Self-running if I provide the hardware, self-hosting if I manage it. Or swap the term, still not sure but there is a difference between you are the one maintaining it and where does it run.

Yourself? In your brain?

Where do you draw the line? Some SaaS acceptable? VPS? Bare metal? Or your own server in your own home?

No, crazy would be owning the router and cables/satellites as well as producing your own power and having a manufacturing plant for hardware capable of running the software.