Robin wrote:

I think the answer is not well defined.

...and then revised that to "1/3 is the better answer, but not obviously so". I wish to agree with both statements, with additional explanation. "Credence" is ambiguous in the question, "What is your credence now for ... heads?" Depending on additional context, that could be a request for Beauty's P(heads | Beauty woken up at least once). Or, it could be a request for her P(heads | Beauty just now woken).

The latter case first: Beauty is offered a bet with some payoff odds, for a few dollars, and she is neither risk-averse nor risk-seeking with regard to such amounts of money. She is to be offered this bet upon each awakening. Nothing else of consequence hinges on the coin flip. In this scenario, Beauty likely interprets "credence" in a manner directly corresponding to betting odds, and her correct answer is 1/3.

Now for the 1/2 case: Beauty knows that her President has decided to launch all-out nuclear war if and only if the coin lands heads. Upon being woken up, her first thought, naturally, is a deep dread at this possibility. How much dread does she feel, in comparison to how she would feel if nuclear war were certain, and in comparison to how she'd feel if it were out of the question? About halfway between.

Since the "offered a bet" scenario is more natural than scenarios where something momentous hangs on the coin flip, 1/3 is probably a "better" answer to the unadorned Sleeping Beauty problem. But even then, your mileage may vary. If you are the kind of person more interested in objective events (the coin flip itself) than the track record of your guesses ("I've just been woken, so I'll say probably tails"), well then, be a halfer. If the opposite, be a thirder.