we want a rigorous, formal explanation of exactly how, when, and why you should or should not stick to your precommitment

Well, if we're designing an AI now, then we have the capability to make a binding precommitment, simply by writing code. And we are still in a position where we can hope for the coin to come down heads. So yes, in that privileged position, we should bind the AI to pay up.

However, to the question as stated, "is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?" I would still answer, "No, you don't achieve your goals/utility by paying up." We're specifically told that the coin has already been flipped. Losing $100 has negative utility, and positive utility isn't on the table.

Alternatively, since it's asking specifically about the decision, I would answer, If you haven't made the decision until after the coin comes down tails, then paying is the wrong decision. Only if you're deciding in advance (when you still hope for heads) can a decision to pay have the best expected value.

Even if deciding in advance, though, it's still not a guaranteed win, but rather a gamble. So I don't see any inconsistency in saying, on the one hand, "You should make a binding precommitment to pay", and on the other hand, "If the coin has already come down tails without a precommitment, you shouldn't pay."

If there were a lottery where the expected value of a ticket was actually positive, and someone comes to you offering to sell you their ticket (at cost price), then it would make sense in advance to buy it, but if you didn't, and then the winners were announced and that ticket didn't win, then buying it no longer makes sense.