For those questioning (rightly) the "self-hosted" description, I suspect its a poorly-worded attempt to clarify there is no third-party requirement (eg. the repo is not cloudflare-owned) beyond your own cloudflare account.

Sometimes these "open source" services are a bit misleading as there is a coupling to the org providing them; I think the OP here just means there's nothing extra needed beyond whatever cloudflare commitment you already have.

"Hosted" is the term that describes an application running on someone else's hardware. As opposed to "self-hosted" which describes an application running on my own hardware. There's no need to invent new terms, just use the existing ones properly and this will prevent any confusion.

"Self-managed" seems the better term.