As pointed out by user LSpice, your definition would be different from the accepted one.

However, it is not a well-constructed definition at all. Indeed, it is possible to have a situation when a function $\phi\colon S \to S$ is $\mathcal{S}$-measurable and $A \in \mathcal{S}$, but $\phi(A)\notin\mathcal{S}$ and hence $\mu(\phi(A))$ has no meaning. E.g., suppose that $S=\{1,2,3\}$, $\mathcal S=\{\emptyset,S,\{1\},\{2,3\}\}$, $\phi(1)=1$, $\phi(2)=\phi(3)=2$, and $A=\{2,3\}$. Then $\phi$ is $\mathcal{S}$-measurable and $A \in \mathcal{S}$, but $\phi(A)=\{2\}\notin\mathcal{S}$.

As pointed out by user LSpice, your definition would be different from the accepted one.

However, it is not a well-constructed definition at all. Indeed, it is possible to have a situation when a function $\phi\colon S \to S$ is $\mathcal{S}$-measurable and $A \in \mathcal{S}$, but $\phi(A)\notin\mathcal{S}$ and hence $\mu(\phi(A))$ has no meaning. For a simplest example, suppose that $S=\{1,2\}$, $\mathcal S=\{\emptyset,S\}$, $\phi(1)=\phi(2)=1$, and $A=S$. Then $\phi$ is $\mathcal{S}$-measurable and $A \in \mathcal{S}$, but $\phi(A)=\{1\}\notin\mathcal{S}$.