I think the issue here is that the two of you are answering subtly different questions - which statement of the problem encourages to arrive at a 'paradox'. In particular, we need to agree that Sleeping Beauty is questioned both times on a result of tails to make progress.
The coin is fair, so the chance of getting heads is obviously 1/2. That's the answer to "What are the odds that a given coin flip came up heads?", but it's not what Sleeping Beauty is being asked when she wakes up.
What SB is being asked is "Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads?" There's an implied conditional present. The coin flip was fair, but the questioning is biased by result - if SB is awake and being asked, the odds are 2/3 that she's in a world where the coin came up tails.
To demonstrate, resort to extreme cases, as you did. If SB isn't woken at all for heads, then the answer to the question "you're awake, how did the coin land?" is tails with 100% certainty. If she's woken up once for heads and 99 times for tails, then the answer to "you're awake, how did the coin land?" is tails with 99% certainty, and so on.
This all hinges on the assertion that SB is questioned on every awakening. If she's only asked on the first awakening (or is asked about the coin flip as a general concept) then the answer is of course 1/2, as you assert.
And herein lies the problem. You say Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads?
But the linked post says (ambiguously) When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?
I don't think your interpretation of the post's question is accurate, but it's worded in vague enough terms that in effect we're arguing about the writer's intent rather than stats.
The coin is fair, so the chance of getting heads is obviously 1/2. That's the answer to "What are the odds that a given coin flip came up heads?", but it's not what Sleeping Beauty is being asked when she wakes up.
What SB is being asked is "Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads?" There's an implied conditional present. The coin flip was fair, but the questioning is biased by result - if SB is awake and being asked, the odds are 2/3 that she's in a world where the coin came up tails.
To demonstrate, resort to extreme cases, as you did. If SB isn't woken at all for heads, then the answer to the question "you're awake, how did the coin land?" is tails with 100% certainty. If she's woken up once for heads and 99 times for tails, then the answer to "you're awake, how did the coin land?" is tails with 99% certainty, and so on.
This all hinges on the assertion that SB is questioned on every awakening. If she's only asked on the first awakening (or is asked about the coin flip as a general concept) then the answer is of course 1/2, as you assert.