I think that Keith Conrad is correct and heights via maxs of valuations are in Weil's "Mordell-Weil" paper. But note that Weil's paper generalized Mordell's work in two ways. First, he extended from $\mathbb{Q}$ to number fields, and second from elliptic curves to abelian varieties (or at least Jacobians). So he would have needed some sort of way of measuring the arithmetic complexity of points on varieties, which he presumably viewed as embedded in projective space.

Note that there's also the more general theory of heights that associates to each divisor $D$ on a variety $X$ (everything defined over $\overline{\mathbb{Q}}$) a height function $h_{X,D}$, or more properly, an equivalence class of functions modulo bounded functions. I believe that this, too, is due to Weil, although this belief comes primarily from the fact that everyone calls them Weil height functions. (If I had to guess, the name "Weil height function" is due to Lang.)

Weil would have fixed a number field $K$ and defined a height function $h_K$ on $\mathbb{P}^n(K)$, and used that to define a height on $X(K)$ for any given embedding $X\hookrightarrow\mathbb{P}^n(K)$. I believe that it was Northcott (*Annals* paper in 1950) who observed that one can define an absolute height on $\mathbb{P}^n(\overline{\mathbb{Q}})$ by taking
$$ h(x) = \frac{1}{[K:\mathbb{Q}]} h_{K}(x) $$
for any number field $K$ such that $x\in\mathbb{P}^n(K)$. Then $h(x)$ is independent of the choice of $K$. So now people often prove theorems saying that "such and such a subset of $\mathbb{P}^n(\overline{\mathbb{Q}})$ is a set of bounded height", which implies in particular that it has only finite intersection with $\mathbb{P}^n(K)$ for any number field $K$, but even more, that it contains only finitely many points defined over all number fields of a given bounded degree. For example, the set of all torsion points on an abelian variety is a set of bounded height.

Note that one can also define height on projective space mimicking the "largest coordinate after clearing denominators" approach, which doesn't involve valuations per se. So, let $a=[a_0,\ldots,a_n]\in\mathbb{P}^n(K)$ for some number field $K$. WLOG, we may assume that all of the $a_i$ are in $O_K$. Then we more or less want to set
$$ h(a) = \sum_{\sigma:K\hookrightarrow\mathbb{C}} \log\max_{0\le i\le n} |\sigma(a_i)|, $$
but there's a problem if the $a_i$ have a nontrivial common factor. Since $O_K$ isn't a PID, we don't have gcds, but we do if we consider ideals. So let
$$ \mathfrak{A} = a_1O_K+a_2O_K+\cdots+a_nO_K $$
be the ideal generated by the coordinates of $a$. Then the correction factor is more-or-less to divide by $\mathfrak{A}$. Precisely, an alternative formula for what we now call the Weil height is
$$ h(a) = \log\left(\frac{1}{N_{K/\mathbb{Q}}\mathfrak{A}}
\prod_{\sigma:K\hookrightarrow\mathbb{C}} \max_{0\le i\le n} |\sigma(a_i)|\right). $$
Now, if you write the norm as a product of prime powers and write the log of the product as a sum, you'll get to usual definition as a sum over all absolute values.