Let $\phi$ = $\forall x \forall y (x \cdot y = y \cdot x)$. It is sometimes said that $\phi$ is "independent" of the axioms of group theory. What does 'independent' mean $\textit{exactly}$? Is it:
(1) That we can find models of group theory $\mathfrak{M}$ and $\mathfrak{N}$ such that $\mathfrak{M} \models \phi$ but $\mathfrak{N} \models \lnot\phi $
Or:
(2) We cannot derive $\phi$ from the axioms of group theory.
Or: is it the case that (1) and (2) are equivalent?
A sentence $\phi$ is independent of $T$ if there are $M,N\models T$ such that $M\models\phi$ and $N\models\neg\phi$. [This is your (1).]
It is a theorem (Gödel completeness theorem) that $\phi$ is independent of $T$ if and only if $\phi$ and $\neg\phi$ cannot be derived $T$ in some (here not defined) syntactic calculus.