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:

• I think it is the simplest form of self-hosting you can do and it is doable by anybody without much technical expertise. For people with little to no technical expertise, it's the perfect gateway to self-hosting. All you need to start is a backup drive.
• For a single person, it's actually the approach that often makes the more sense.
• And even for technical people, sometimes you just don't want to deploy and maintain yet-another-service.
• And finally, you can still access your data when you're offline.

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.