I'm sure somebody will put up the stacky answer to this question, but let me try to give a more down to earth one.

The point is that projective space is a defined using a functor Proj. Proj has the pleasant property that if X is any scheme, and L is a line bundle, then there is a canonical map from $X\_0$ to $\mathrm{Proj}(\oplus\_{n\geq 0}\Gamma(X;L^{\otimes n}))$, where $X_0$ is the subset of $X$ where at least one section of $L^{\otimes n}$ is nonvanishing for some n. This doesn't use any choices or algebraically closed base; for any section of $L^{\otimes n}$, we just take the map to the affinization of the non-vanishing set and glue all of these together.

So, if $L$ is globally generated, we have a map from $X$ to $\mathrm{Proj}(\oplus_{n\geq 0}\Gamma(X;L^{\otimes n}))$, and this is the universal polarized projective variety such that the pullback of $\mathcal{O}(1)$ is $L$. Now, we just need to find maps of this variety to $\mathbb{P}^n$ that give the right line bundle.

So we need to think about what maps between two Projs (preserving the line bundle) are, and that's quite simple, it's maps of graded rings the other way which hit all non-irrelevant ideals.

Since $\mathbb{P}^n$ is Proj of a polynomial ring, a map of a Proj to projective space picking $n$ degree 1 sections who generate an algebra that hits all non-irrelevant ideals, which is exactly the description you gave.

I'm sure somebody will put up the stacky answer to this question, but let me try to give a more down to earth one.

The point is that projective space is a defined using a functor $\mathrm{Proj}$. $\mathrm{Proj}$ has the pleasant property that if $X$ is any scheme, and $L$ is a line bundle, then there is a canonical map from $X_0$ to $\mathrm{Proj}(\oplus_{n\geq 0}\Gamma(X;L^{\otimes n}))$, where $X_0$ is the subset of $X$ where at least one section of $L^{\otimes n}$ is nonvanishing for some n. This doesn't use any choices or algebraically closed base; for any section of $L^{\otimes n}$, we just take the map to the affinization of the non-vanishing set and glue all of these together.

So, if $L$ is globally generated, we have a map from $X$ to $\mathrm{Proj}(\oplus_{n\geq 0}\Gamma(X;L^{\otimes n}))$, and this is the universal polarized projective variety such that the pullback of $\mathcal{O}(1)$ is $L$. Now, we just need to find maps of this variety to $\mathbb{P}^n$ that give the right line bundle.

So we need to think about what maps between two Projs (preserving the line bundle) are, and that's quite simple, it's maps of graded rings the other way which hit all non-irrelevant ideals.

Since $\mathbb{P}^n$ is Proj of a polynomial ring, a map of a Proj to projective space picking $n$ degree 1 sections who generate an algebra that hits all non-irrelevant ideals, which is exactly the description you gave.