> If these two distinct constructs were indeed different ways to define the same relationship, the second paragraph would say "^3 and ~3 are both ways of saying the third parent of the current commit, or the parent's parent's parent."

They're not the same thing. Merge commits can have multiple parents, and HEAD^3 refers to the third parent. If HEAD is not a merge commit, then HEAD^3 doesn't refer to anything. HEAD~3, by contrast, refers to following the parent's parent's parent, independent of how many parents any of the commits in question had.