Weil height vs Moriwaki height

Question

Let $X$ be a projective veriety over a number field. After fixing an embedding into $\mathbb P^n$ (i.e. a very ample line bundle $L$), one can define the Weil height $\hat h_{L}$ by restriction of the naive height on $\mathbb P^n$ to $X$.

Around 2000 Moriwaki defined a different notion of height in the following way: fix a couple $(X,L)$ where $X$ is as above and $L$ is a line bundle on $X$. Moreover choose a model $(\mathcal X,\overline{\mathcal L})$ where $\mathcal X$ is an arithmetic variety over $O_K$ and $\overline{\mathcal L}$ is a hermitian line bundle on $\mathcal X$ extending $L$. Then by means of Arakelov intersection theory (actually Gillet-Soule' theory) one can define the height function $h_{(\mathcal X,\overline{\mathcal L})}$.

What is the advantage of working with the Moriwaki height $h$ instead of $\hat h$? Why Weil height was not enough?

As far as I understand, if one fixes $L=O(1)$ and operates with the Fubini-Study metric restricted to $X$ it is possible to recover $\widehat h$ as a special case of the Moriwaki height... (this is shown in chapter 9 of Moriwaki's book on Arakelov Geometry)

Why do we need the whole machinery of arithmetic intersection theory in order to obtain "just" a generalisation of the Weil height?

Answer 1

One can define a height with respect to a special metric so that some particular useful equation involving the height holds exactly, instead of approximately as for the Weil height.

For example, one can define the canonical height using a metrized line bundle. This puts it on the same footing as the Weil height and not defined only as a limit of Weil heights composed with various maps.

More substantially, the Faltings height was defined this way, using a metrized line bundle on the moduli stack of abelian varieties. The invariance of the Faltings height under isogenies associated to p-divisible groups was crucial in Faltings' proof of the Mordell and Shafarevich conjectures.

Edit: Probably the answer to the second question is one doesn't need the full machinery of arithmetic intersection theory, i.e. one can define the height associated to a metrized line bundle without invoking all the techniques needed to handle arithmetic intersection theory in full generality. One just needs to take a section and add up its valuation at each place.