But doesn't shareware refer to software that is distributed freely, playable (maybe with limitations on how far you can play into the game, or how long you can play it for free) but it's generally a proprietary game that is distributed through this model? I may not have a perfect grasp on the precise meaning of shareware.

I see chess more like abandonware - if someone developed a proto-chess game and didn't assert their ownership over the IP (recognising that this happened before copyright and IP were understood concepts), doesn't that make it effectively free to play, noting you of course need a board to play?

Maybe an ancient and highly modded board game doesn't translate that well to a software/copyright analogy. Also you lost me on your comparison between Mario Party (I think you mean only one person needs to own the game) and the first levels of Doom (which are more like a demo). I don't see either of these as shareware, though I guess a freely playable demo is a form of shareware with my understanding of the term above!