Not that they are necessarily using multiple definitions, but the common definitions themselves do not specify an exact range in quantity in which a cluster of things could be considered a heap. Two people could disagree about whether something is a heap despite using the same vague definition of "heap". [...]

Ahhh, I think I understand. Yes, that makes sense.

Doesn't the introduction of a simplification itself give a term some vagueness, because then you don't know the details of the relevant characteristic, just a qualitative judgment?

Yes, although (as you suggest) that's a different kind of vagueness: vagueness as a definition's inclusiveness vs. vagueness as uncertainty about the definition itself.