Denote by $\mathbb{P}^n:= \mathrm{Proj} \, \mathbb{Z}[X_0, \ldots ,X_n]$. I have read the following somewhere and don't understand it:
For a scheme $S$ we choose coordinates $z_i$ with $0 \leq i \leq n$ for $\mathbb{P}^n(S):= \mathrm{Mor}(S,\mathbb{P}^n)$.
How is this done explicitly? Does it have something to do with the natural transformation between $\mathrm{Hom}_S(X, \mathbb{P}^{n+1})$ and isomorphism classes of tuples $(\mathcal{L}, s_0, \ldots , s_n)$ where $\mathcal{L}$ is a line bundle and the $s_i \in \mathcal{L}(X)$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$?
You have already given the correct answer.
Indeed, any morphism
$$\varphi : X\longrightarrow \mathbf{P}^n=\mathrm{Proj}(\mathbb{Z}[X_0,\ldots,X_n])$$
is associated to a complete linear system $|\mathscr{L}|$ in the following way. Let $\mathscr{L}:=\varphi^* \mathscr{O}_{\mathbf{P}^n}(1)$ be the pullback of the sheaf of linear forms. Then since $\mathscr{O}_{\mathbf{P}^n}(1)$ is globally generated by global sections $X_0,\ldots,X_n$ (which correspond to a choice of coordinates on $\mathbb{P}^n$, it follows that $\mathscr L$ is globally generated by global sections $\sigma_j:=\varphi^* X_j$. This means that, for every $p\in X$, we have $$\varphi(p)=[X_0(\varphi(p)):\cdots :X_n(\varphi(p))]=[\sigma_0(p):\cdots :\sigma_n(p)]$$ So the sections $\sigma_0,\ldots,\sigma_n$ act as projective coordinates of $\varphi$.