I'm coming around to the 1/2 point of view, from an initial intuition that 1/3 made most sense, but that it mostly depended on what you took "credence" to mean.

My main new insight is that the description of the set-up deliberately introduces confusion, it makes it seem as if there are two very different situations of "background knowledge", X being "a coin flip" and X' being "a coin flip plus drugs and amnesia". So that P(heads|X) may not equal P(heads|X').

This comment makes the strongest case I've seen that the difference is one that makes no difference. Yes, the setup description strongly steers us in the direction of taking "credence" to refer to the number of times my guess about the event is right. If Beauty got a candy bar each time she guessed right she'd want to guess tails. But on reflection what seems to matter in terms of being well-calibrated on the original question is how many distinct events I'm right about.

Take away the drug and amnesia, and suppose instead that Beauty is just absent-minded. On Tuesday when you ask her, she says: "Oh crap, you asked me that yesterday, and I said 1/2. But I totally forget if you were going to ask me twice on tails or on heads. You'd think with all they wrote about this setup I'd remember it. I've no idea really, I'll have to go with 1/2 again. Should be 1 for one or the other, but what can I say, I just forget."

I'm less than impressed with the signal-to-noise ratio in the recent discussion, in particular the back-and-forth between neq1 and timtyler. As a general observation backed by experience in other fora, the more people are responding in real time to a controversial topic, the less likely they are to be contributing useful insights.

I'm not ruling out changing my mind again. :)