Two definitions of ergodicity
One can show that the first definition is equivalent to the following one:
A measure preserving transformation $T$ is ergodic if for every measurable set $A$ with $\mu(A\Delta T^{-1}(A))=0$, $\mu(A)=0$ or $\mu(A^c)=0$.
This comes from the fact that for any such $A$ there is a set $B$ s.t. $T^{-1}(B)=B$ and $\mu(B\Delta A)=0$.
Now, in the second definition $E\supset T^{-1}(E)$, which implies that $\mu(E\Delta T^{-1}(E))=0$ because $\mu(E)=\mu(T^{-1}(E))$, and so it is equivalent to the first definition.