Yes, this is the modern statement of Weil's theorem of decomposition. For a more recent exposition see 2.7.15 in Bombieri and Gubler's Heights in Diophantine Geometry. It is a basic component of the theory of heights.
If you look for a specific application of the theorem and its point of view, you should be aware of Bombieri's paper [On Weil's "Theoreme de decomposition," Amer. J. Math., 1983]. There, Bombieri employs the theory of heights (Weil's theorem of decomposition and a theorem of Neron, cf. 9.3.10 in Bombieri-Gubler), to extend a result of Sprindzhuk on the Hilbert irreducibility theorem and deduce from it a generalization of an old theorem of Runge stating the finiteness of solutions $(x,y) \in \mathbb{Z} \times \mathbb{Q}$ to $G(x,y) = 0$ for an irreducible $G \in \mathbb{Z}[x,y]$ whose leading homogeneous part is not proportional to a power of an irreducible polynomial.
Roughly, Bombieri's result is that if $f : C \to \mathbb{P}^1$ is a morphism from a curve over a number field $K$, and $P \in C(\bar{K}) \setminus f^{-1}(\infty)$, then as the place $v$ of $K$ varies, each pole of $f$ is $v$-adically approached by $P$ with frequency proportional to the order of that pole. (Or, taking multiplicities into account, all poles of $f$ are approached with the same frequency.) This is very much in the spirit of Weil's theorem of decomposition. A modern exposition of Bombieri's result and its consequent generalization of Runge's theorem is presented in the chapter 9 on Neron-Tate heights in Bombieri and Gubler's book, where it is used to give an essentially algebro-geometric proof of the Hilbert irreducibility theorem.