Two ways of smooth dragging a vector on a Lie group

Question

I know two ways of dragging a vector tangent to the Lie group $G$ from $T_gG$ to $T_eG$:

1. Using Maurer-Cartan form $\theta$ which acts as a push-forward of the left group translation by $g^{-1}$: $$\theta(X|_g) = L_{g^{-1}*}(X|_g)$$

2. Using Lie transportation. I guess, nothing stops me form finding such vector field $Y$ that dragging $X$ along $Y$ would move $X$ to $e$.

Can these two be reduced to each other?