Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $L$ be a line bundle on $A$. Then a Néron–Tate height $\hat h_L$ can be defined by taking a model $\mathcal A$ of $A$ (over the ring of integers of a field of definition of A) and a hermitian line bundle $\mathcal L$ on $\mathcal A$ that "extends" $L$. The definition of $\hat h_L$ then involves some arithmetic intersection theory on $\mathcal A$.
Now consider $A=\mathbb G^n_m(\mathbb{\overline Q}) $ (in this case $A$ is a group scheme). A very quick way of definining “the” canonical height is the following:
$$ \hat h(x_1,\dotsc, x_n)=h(x_1)+\dotsb+ h(x_n)\quad (\ast) $$
where $h$ is the usual Weil height on $\overline{\mathbb Q}$.
How are these two notions of height related? In particular what line bundle is hidden in the second "quick" definition?
Another obvious definition of height is by embedding $\mathbb G^n_m(\mathbb{\overline Q}) $ inside $\mathbb P^ {n}$ and then taking the restricted Weil height, but I suppose that in this case we don't have the nice properties of the Néron–Tate height.
edit: I suppose that $(\ast)$ is just a renormalisation of the standard Weil height on $\mathbb P^n$ (i.e. they differ by O(1)) but there is the advantage that $\widehat h$ is $0$ only on (componentiwise) torsion points