That's what happens in it for the former, not so much what it's about. As for identity, that's an example of an assigned label.
Two heads, specifically, though it supposedly was a coin with two heads when the flip began, even if it is no longer validly identified as a coin with two heads when it lands. Frankly, it's doable in a few ways, but the entire two headed coin tangent is rather pointless and little more than an empty distraction, in my opinion.
If you stretch things a bit, it can, but it's likely not the best thing to use. Fudbucker has demonstrated that he has a habit of using terms in not quite proper ways, though, so this really isn't a surprise.
As noted, there are a couple tricks that could be used, potentially. If you're assuming that no tricks were used, which I don't think was ever stated to be either the case or part of the description, you're likely right that there's no prior probability, though. Either way, invoking the chance of something completely unexpected happening isn't specifically wrong to do. It's just fairly pointless on a practical level, much like it's not wrong to acknowledge the usually vanishingly small chance that a flipped coin will settle on its edge, rather than either side for whatever reason, but not all that useful to take into account. It's the kind of thing that may as well pass without comment.