One thing that seems to be confusing you is that there is no "the moment map." There often several different moment maps for the same action, and as Jose says, sometimes none (though there always will be if you're willing to replace your group by a central extension).
On the other hand, your questions seem to really be
If X is a vector field, are the conditions
- $X$ preserves $\omega$ and
- $i_X\omega=\omega(X,-)$ is an exact 1-form
equivalent?"
The answer is yes. The preservation of $\omega$ by $X$ is the same as saying that the Lie derivative $\mathcal{L}_X\omega=0$. By Cartan's magic formula,
$$\mathcal{L}_X\omega=d(i_X\omega)+i_X(d\omega)=d(i_X\omega)$$ since the second term is zero by the exactness of $\omega$.