I recently learnt the notion of "Hodge bundle", primarily for families of abelian varieties. This is usually defined as follows: Let $\pi:X\to S$ be an abelian scheme and $e:S\to X$ be its zero section. Then the Hodge bundle is the pullback bundle $$\omega=\omega_{X/S}:=e^{\ast}\Omega^1_{X/S}$$ where $\Omega^1_{X/S}$ is the sheaf of relative differential $1$-forms on $X$. It seems some authors also call the pullback $e^{\ast}\Omega^g_{X/S}$ of the top wedge of $\Omega_{X/S}^1$ the "Hodge bundle" (where $g$ is the relative dimension of $X/S$).
However, I also see a definition of Hodge bundle using the pushfoward, instead of the pullback, of the sheaf of relative differentials. For example, in the ELSV formula (one can consult its Wikipedia page for more details.), the Hodge bundle $\omega$ on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points is $$\omega:= \pi_{\ast}\omega_g$$where $\omega_g$ is the relative dualizing sheaf on the universal stable curve $\pi:\mathcal{C}_{g}\to \overline{\mathcal{M}}_{g,n}$.
My question is: what is the relation between the two notions of Hodge bundles? More precisely, if $\pi:X\to S$ is a "nice" family of schemes (e.g., abelian varieties, curves, etc.) of relative dimension $g$, and $e:S\to X$ is the section, then what are the relations between $e^{\ast}\Omega^1_{X/S}$, $\pi_{\ast}\Omega^1_{X/S}$, and their higher wedges?
The point here is that for an abelian variety $A$ over a field, any regular differential form on $A$ has to be translation-invariant; hence a differential form is uniquely determined by its fibre at the origin. Extending this to families $A / S$, we get that the evident map of sheaves $\pi_* (\Omega^1_{A / S}) \to e^*(\Omega^1_{A/S})$ is an isomorphism (and we can take the $g$-th wedge power if we want).
Some authors prefer to use one or the other of the two notations, $e^*$ vs $\pi_*$, depending on taste. For abelian varieties it genuinely doesn't matter. However, one should be aware that the two things no longer match for semiabelian varieties (arising from degenerating families of abelian varieties a la Mumford--Tate), in which case $\pi_* \Omega^1$ may not even be coherent (because the bad fibres aren't proper), so $e^*$ is usually the better object.