My point was not that the situation is analogous to PD (the waiter doesn't play, it's a one player decision, not a two player game).

Not true (that it is single player game), but this is tangential.

Well, it seems obvious to me that this is a one player game, so for me it's not tangential, it's very important for me to correct the error on this. As I see it, the only decision here is whether to tip, and this decision is made by the customer. Where is the other player, what is its action?

make all the other relevant participants in the game rational agents rather than humans with known cultural programming. In the actual problem you have no more reason to (act as if you) believe you are (or could be a priori) the waiter than to believe you are the cow that you are served or the fork you use to eat the slaughtered, barbecued cow.

Rationality of the other participants is only relevant to the choice of their actions, and no actions of the waiter are involved in this thought experiment (as far as I can see or stipulate in my interpretation). So indeed the waiter is analogous to a cow in this respect, as a cow's inability to make good decisions is equally irrelevant. It's value of personal prosperity that the hypotheticals compare. The distinction I'm drawing attention to is how you care about yourself vs. how you could counterfactually care about the waiter if you were the waiter (or a cow if you were the cow), not how you make decisions yourself vs. how the waiter (or a cow) makes decisions.

It is enlightening to work on better understanding of what given considerations actually imply.

These considerations don't apply.

That's exactly the question I'm considering. I'm not sure if they apply or not, or what they suggest if they do, I don't know how to think about this problem so as to see this clearly. You insist that they don't, but that doesn't help me if you don't help me understand how they don't.

One sense of "applying" for an idea is when you can make novel conclusions about a problem by making an analogy with the idea. Since I'm not making novel conclusions (any conclusions!), in this sense the idea indeed doesn't apply. What I am insisting on is that my state of knowledge doesn't justify certainty in the decision in question, and I'm skeptical of certainty in others being justified.