Re: "If we are being asked about the probability of our coin flip associated with this iteration of the experiment, then the answer is 1/2. If we are being asked about the probability of the coin flip associated with this particular awakening, then it must be 1/3."

What is actually asked at each awakening is:

"What is your credence now for the proposition that our coin landed heads?"

I figure that makes the answer 1/3 - and not 1/2.

If the question had been: "What is your credence that this is the last time you awaken and our coin landed heads-up?"

...then the answer would have been 1/2.

...but that wasn't the question that was asked.