Ito's formula/lemma is like the chain rule from calculus. It is a generalization, in that it uses a second order Taylor series expansion, whereas the chain rule only needs a first order expansion. Anyway, I think (2) is a reflection of this fact, and how the chain rule lets us compute dynamics of a derived process.

I sort of disagree with (1), since Ito's lemma is most naturally applied to ~martingales, of which Brownian Motion is an important special case.