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.

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 "no", but it's "yes" if you replace "exact" by "closed".  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 closedness of $\omega$.