I think the problem is in the phrasing of the question: When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

That's not a precise probability question.

More significantly the question (whatever it might be) is potentially asked more than once.

So, if the question is "what is the probability that the coin-flip was Heads?" (which I think it the most likely interpretation) then it's 1/2

But if the question is "what is the probability that you are being woken up because the coin was Heads" then it's 1/3

Asking me what I "should believe about the outcome" is (intentionally, I suppose) blurring the question so that it disguises the fact that there are actually 2 different questions in play.