Is free abelian group a "free" abelian group or "free abelian group"?

Question

Let $G$ be an abelian group.

What does it mean that $G$ is a free abelian group? Does this mean that $G$ is a free group or a free-$\mathbb{Z}$-module with the operation $n•a=a+...+a (n-times)$?

Or are they equivalent?

EDIT:

I just realized that no abelian group is free-group since every nonzero element does not have a unique canonical form under basis.

Answer 1

There are two free groups that are abelian! Namely: $\mathbb{1}$ and $\mathbb{Z}$.

A free group on at least two generators cannot be abelian, because the nonabelian group $S_3$ is its quotient (as it can be generated by two elements).