Is there a common source y'all are drawing on that proposes games not to require player-characters? I'd like to read it.

A quick search on zero player games should give you this: http://www.jesperjuul.net/text/zeroplayergames/

Unfortunately I have never found a book focus just on zero player games, it doesn't mean thy don't exist though.

Usually zero player games topics can be found in AI books, there are also the studies on zero player games scattered over the internet.

A game has to be played voluntarily.

This is interesting. When scientist look at animals who can play games, the animals must play voluntarily.

So a snail who is raced against others isn't playing a game.

When humans are forced to play games it's still considered that thy are indeed playing, many horror movies are made about players being forced to play a game.

I am not a hundred percent sure of this but I think it's because when the snails are placed thy have no idea what is happening. Then if you did the same with people thy would never race unless thy have a motive.

So you have to force them to play instead of making the game around them? Then what about the fish that played Pokemon, did he really play?

For rigor, you should perhaps try to pin down the larger question that motivates this terminological one. Why is it important what English speakers call "games"? Not all languages have this word, or they have a word that refers to a different set of phenomena. Why does it matter what set of things English speakers pick out as games? (And note that this changes over time. Lots of people were hesitant to describe SimCity as a game when it was released; now, due in part to its success, people term that software category 'games' without hesitation.)

(Have you ever sat around and debated, with a sufficiently large group of people, the necessary and sufficient condition for "mug", or "chair", or "jacket"? Presumably you haven't --you should, it's weirdly contentious -- but you'd probably agree that it doesn't actually matter what set of drinking utensils counts as "mug". We will continue to manufacture and drink out of drinking vessels regardless of whether anyone agrees what "mug" is.)

I note this because competing definitions aren't always trying to do this -- when they say "A game has win conditions", they don't mean "Everything that people currently describe as a 'game' has win conditions"; ascribing that to the authors violates the principle of charity, as if you think they're not seeing obvious counterexamples. Rather, they're saying "The development of the field needs technical definitions, and we're better off calling a narrower set of phenomena 'game' than what ordinary people call 'game'." (In part this is so that we can communicate technically without spending all of our time in "mug" arguments.) So if you want to shift the terms of the debate back to usage, you should be clear that you want to do this and why it's important, and if your goal is something else than characterizing usage, you should be clear about that, too, so that readers can judge whether your proposed definition fulfills your goals in proposing it. You throw out words like "misguided" but you don't characterize what guides their definitions OR yours.

As a different note (and many definitions of games face this), you define a game in terms of ideas that themselves assume a definition of game already. I like your distinction between player and player-character, but when you use these, and "rule", you already mean "of the sort we find in games", but you then define "game" in terms of these. You can do this, of course, if your goal is to do something else than define "games" -- for example, if you really just want to make a typology of rules, do go ahead -- but it is mis-framing the debate again to say that other definitions are "redundant" and can be reduced to yours if your definition is circular.