It is “hosted” on your workstation. There is no need for a server-client relationship for self-hosting.
By requiring a server-client relationship, you’re making self-hosting uselessly hard to deploy and enforce a very specific design when others (P2P, file sync, etc) can solve the same problems more efficiently. For example, in my specific case, with Paperwork + Nextcloud file sync, my documents are distributed on all my workstations and always available even if offline. Another example is Syncthing which IMO fits the bill for self-hosting, but doesn’t fit your definition of self-hosted.
Yes I would count this game as self-hosted (as long as you don’t need a third-party service to start it). And yes I agree it is a pretty wide definition. But at the same time, I really think there are a lot of good reasons to not dismiss it:
To be honest, when it comes to self-hosting, I can’t shake this feeling that a lot of people are dismissing desktop apps immediately just because they are not cool nor hype anymore.
Regarding Syncthing, if I’m not mistaken, the Web UI can be opened to the network (most likely for headless servers) but by default it is only reachable through the loopback.
Regarding OP, for me, it wasn’t entirely clear at first whether they wanted network access or not. They clarified it later in comments.