Understanding free product groups

Question

I am trying to understand free product groups and I considered the group $G=\langle a,b|a^2b^{-3}\rangle$. As in this group $a^2=b^3$, it seems like every element of G can be written as $b^{n_1}ab^{n_2}a\ldots$, as every $a$ to a power greater than 2 "disappears" into a power of b. Hence i feel like $G$ is the free product $\mathbb{Z} \star C_2$ where $C_2$ is the cyclic group of order 2. It appears that my intuition is probably false as a similar reasoning would lead to say that it is also the free group $\mathbb{Z}\star C_3$. Can someone explain me why my intuition is false ?