Dialogue 2 and dialogue 3 as you phrased them are equivalent, but they both omit a significant aspect of the original discussion -- that Omega promises that if the coin had come up tails, it would have offered the same bet (which would now have been a winning one for you).

Mmm... I implied it by saying in the first paragraph that Omega is reliable and that's common knowledge, but it's true that the wording could have been much clearer. I wonder if an edit would do more harm than good.

Your scenarios, as stated, leave it unclear whether the bet is offered because it came up heads. So the possibility is left open that Omega only offer bets when he knows that the coin came up heads.

That's on purpose, and is exactly the information that the agent doesn't have in one-shot counterfactual mugging.

Taking the scenario where we know that Omega would have offered the bet regardless of what the coin-toss was: that effectively means that, in this type of decisions, statistically speaking, agents are favoured who exhibit some sort of "timelessness" in their decision theory, agents are favoured who do not update in the sense that a CDT agent would update.

So to have that "winning decision theory" (which is winning overall, not in individual cases), we must be agents who do not update in this manner.

That is all well estabilished, I think. Yet:

The problem people tend to have with this is that they seem to assume a winning decision theory to be one which maximizes the expected utility of each of any individual decisions as if they're logically independent from each other, but in reality we want a decision theory that maximizes the expected summed utility over the whole life-length of the decision theory.

A DT though that would accept only winning counterfactual but behaves in every other occasion like a TDT agent is even better. My problem is understanding if the rule proposed is consistent with this desiderata, or would turn a TDT back into a CDT.