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No, it isn't. Being woken up is evidence for tails. So if they ask you after waking you up, you have additional evidence that you did not have when they asked you before the experiment.
(And if your reply is "well, didn't you know in advance that you would be awoken?" the answer is that "being awake" and "knowing that you will be awake" don't provide the same evidence, because they are distributed among the outcomes differently.)
Note the phrasing:
Not:
The former is a question about a reality that continues to exist outside of our personal observations. The latter is a description of assumptions you can make while biased under this or that frame that limit your observational abilities. These are different questions and have different answers. Again, as described, the gambling case makes the practical side of this very clear, but this shouldn't blind us to the absolute perspective.
As for why this matters: imagine that the researchers tell you what they flipped before you go to sleep the first time. This is the analogue to real-world scenarios, where there always is a driving factor of variance, but we rarely get a privileged peek behind the curtain as to what it is. Describing this or the other real world event as probabilistic is helpful primarily for placing ourselves within our own information-blind reality, but if you are able to get a real look at the coin, everything changes. That's why it's important to understand the odds, of course, but also to understand there's something behind them. If you at all aspire to a scientific understanding of your situation, you must not be thinking about the odds, you must be thinking about getting a look at that coin.
Well, ok, but you chose that ambiguous phrasing. The Wikipedia article has two different statements of the problem, neither of which is unclear. You have to be very careful with your wording (as you were) to make it a misleading question that sounds like it's asking about a result but is actually, uh, about a "reality that continues to exist".
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In that case I would agree that the problem is phrased ambiguously. The per experiment probability is 50% and the per-awakening probability is 1/3.
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