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On the first point, you're right that it is possible to ask this question. I suppose I exaggerated what I was trying to say. The issue I think is language tense. If you ask in the progressive tense "what are the odds of this happening, then you are asking someone about repeated probabilities. "If I, knowing nothing, get on a plane, what are the odds of A and B happen simultaneously?" The correct answer would be to compute the probability of A, the probability of B, and then multiply them together. Because you're not asking about whether this happened in the real world, but about whether it could/would happen in general.
If you ask in the past tense "what are the odds that this happened, this is a question about the world. This is actually the question "What are the odds that this thing happened, conditional on everything you know right now, including me asking you this question?" It is not a question about general repeated probabilities, because that's not how verb tenses work. It's past tense. You could convert it into a question about repeated probabilities (which you might need to if you are a frequentist), but if you did it would translate into "What are the odds of this thing happening conditional on you finding yourself in a mathematically analogous situation to the one you find yourself in now." If you ask me the probability that you yourself were sandwiched between Avril Lavigne and Justin Bieber on a flight I'm not going to compute the probability of them being on flights, I'm going to say ~0% because if that had actually happened you would have phrased it very differently when using it as an example.
You're also right that I mangled my example while editing. The example is supposed to create a scenario where there's a 50% everything is normal (we flip one coin and it's heads) a 50% chance we have a flaky bookie (who in turn has a 50% chance of reneging on his bet). The point is not that the example is "contrived", the point is that it detaches betting odds from probabilities because the payouts are distorted. Consider a friend who, on a first roll, fumbles his dice and drops them clumsily. If the result is a 1 he says it doesn't count and rerolls them properly, keeping the result no matter what. But if the fumbled roll is good he keeps it. If this were a consistent pattern you would be on his dice differently than 1/6 per side, because you're not betting on the probability that a die rolls a certain number in a vacuum, but the probability that a certain number is kept in the end.
When sleeping Beauty wakes and makes a bet, there's a chance your version 2 is going to discard her bet and roll again, only accepting her bet if she wakes up and makes the same bet again the next day. If she always bets on "heads" she will be wrong 2/3 of the time she says it, but lose money 1 time and gain money 1 time. You might as well never wake her up on Tuesday at all because you're essentially taking bets on Monday in both cases and then ignoring her Tuesday answer unless it conflict with Monday. The probability you're actually getting here is "Conditional on me asking you this question and this being a day when your answer actually matters for betting purposes, what is the probability of it being heads?" which is a very very different question from "what is your belief that the coin is heads right now?" which is what she's actually asked in the original question.
I think a more straightforward way to notice that this scenario detaches P(heads|you just woke up) from the optimal betting strategy is to compare it to the following scenario:
Some researchers flip a coin without showing you the result. On Monday, they interview you about the coin and ask you to make bets about its status. Then, on Tuesday, if the result was tails, the researchers play the videotape of your interview from Monday and perform all your bets a second time on your behalf.
Here, your belief that the coin landed on tails should clearly be 0.5 even given the condition that you're currently being interviewed. But if you make any bets, you need to keep in mind that they'll be executed twice in the tails condition. The optimal strategy is the same as in the original Sleeping Beauty problem, since that problem supposes that you were going to do the same thing on both days anyway. (That strategy is not as straightforward as "assign probability 0.6667 to tails" if you can bet things like "all the money currently in my checking account" rather than just fixed dollar amounts.)
So within the problem, the concept of "credence" is not as broadly applicable as it normally is; the conditional probability is different from the optimal betting odds (and those odds themselves differ based on details of the bet). You can either stick to the conditional-probability definition, say that the odds are 0.5 (0.6667 in the original problem), and not use that value for any practical purpose. Or you can say "I think there is a 50% chance that the coin is tails, and if that is the case any actions I take will happen twice", which is a more useful fact to know when strategizing.
I think you detached them in the opposite way here. In the original problem both the conditional probability and optimal betting odds are 0.6667. In /u/4bpp version (and the version I attempted to describe) the conditional probability is still 0.6667 but the optimal betting odds go to 0.5. In your version the conditional probability is 0.5 and the optimal betting odds are 0.6667. You are correct that this is an easier way to describe how betting odds and conditional probabilities can detach.
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