Then it occurred to me that you made the same assumption as in my post shortly prior to yours: a priviledge of "ME" to switch, barring the others.
I think this pinpoints one of the confusions that's muddying up this discussion. Under the Flip-Flop rules as they were presented, the Winning Flip is determined before people switch, and the Winning Flip doesn't change based on how people switch. In that scenario, my table is correct, and there is no paradox.
We can also consider the variant in which the Winning Flip is determined after people decide whether or not to switch. But that game is functionally identical to the game where there is no coin-toss at all - everyone just freely chooses Heads or Tails, then the Winning Flip is determined and the winners are paid. Flipping a coin, looking at it, and then deciding whether or not to switch it is identical to simply picking heads or tails! The coin-flips only matter in the first variant, where they determine the Winning Flip *before* people make their choices.
In this variant, it doesn't matter whether you switch or not (i.e. whether you choose heads or tails) - you are more likely to lose than win. We can use the same 3-player table we've been discussing to see that there are eight possible outcomes, and you only win in two of them. Once again, there's no paradox, although you might *feel* like there is one. You might reason that the Winning Flip is equally likely to be heads or tails, so no matter which one you pick, your odds of winning will be 50/50. What's missing from this logic is the recognition that no matter what you pick, your choice will automatically decrease the chances of that side being in the minority.