Does a Galois group being $S_3$ correspond to the extension being the splitting field of a cubic?

Yes. To see this, note that if $K/F$ is a Galois extension, then by the Fundamental Theorem of Galois Theory there is an intermediate subfield $L$ with $[L : F] = 3$, corresponding to the fixed field of $\langle (1,2) \rangle$. By the Primitive element theorem, which in particular holds for any Galois extension, $L = F[\theta]$ for some $\theta \in L$. Let $p(x)$ denote the minimal polynomial of $\theta$. Then $degree(p(x)) = 3$.

Now let $r \in K$ be another root of $p(x)$. If $p \in L$, then $L/F$ would also be a Galois extension--which contradicts the fact that the group $\langle (1,2) \rangle $ is not a normal subgroup of $S_3$. Thus claim $r \notin L$, which gives that $K$ is the splitting field of $p(x)$.