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His definition of rare is an artifact of the notation where board state requires disambiguation, as in it includes externalities.

I feel like the question remains unanswered

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The problem is that there must be hundreds of ways to describe "a move" from the complete board state before and after to the distance that the piece moved (moving moving a bishop and queen 2 diagonally is the same move).

So while I agree that the notation is pretty arbitrary and puts lots of emphasis on how implicitly a move can be recorded I don't think it is fundamentally worse than any other definition. Yes, personally I would have picked something more directly tied to the game than if the notation requires more disambiguation, but I don't think it really makes the video any less interesting or the result and less useful (probably no use either way).

Agreed. Just the notion of double disambiguation is meaningless to a chess player who doesn’t use the notation. This is fundamentally not about chess. But about one particular way that certain people use to write down chess moves.

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