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Wouldn't this definition be easier:
A board play is the same as another board play, if
1) both start from the same distribution of stones, and
2) both end with the same distribution of stones.
I am not sure if this is sufficiently precise, plus someone might insist that you would have to define "distribution of stones" (or "position" or whatever), but it seems easier to grasp. I think Robert Jasiek uses a similar definition in one of his ko rules, so you could look there for another (perhaps better) wording.
Well, it may be easier, but I do not think that it accords with Ing's approach. When Ing says that a disturbing ko may be played only once, I don't think that he is referring to a whole board position (which is what I take you to mean by a distribution of stones). He means something more local.