> even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.

I think this is practically fixed by Robinson's NSA when it's combined with big/little O notation:

  δy = f'(x) δx + o(δx)

A (nonstandard real) quantity is o(δx) when it's infinitesimal relative to δx, i.e. s ∈ o(δx) whenever s/δx is infinitesimal. So δx² ∈ o(δx) but δx ∉ o(δx).