Does this make sense as an alternative definition for a finitely-generated field extension?:
A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$.
And similarly, $K \subseteq F$ is a simple extension if and only if there is a ring epimorphism $K[t] \twoheadrightarrow F$.
Epimorphism here is used in the categorical sense, and is in general not going to be surjective.
I'd just like verification on this. It is subtly implied in Aluffi's Algebra Chapter 0 in chapter VIII Remark 1.14. The proof of the Nullstellensatz in section 2.2 relies fundamentally on the implication that if $F$ is a finite-type $K$-algebra, then $F$ must also be finitely-generated as a field extension.
However, the definition for simple and finitely-generated field extensions (in Aluffi, as in other places I've checked) is unsatisfying to me. Namely, to show $K \subseteq F$ is finitely-genreated (or simple) you need to first exhibit a field extension on your own, pick appropriate elements to adjoin, and then show your extension is isomorphic to the "smallest field containing the adjoined elements".
However, having the epimorphism definition would make it easier to reason by diagrams, and it seems like it should trivially imply that fields $F$ which are finite-type $K$-algebras are finitely-generated ring extensions just by "forgetting" you're working in $K$-Alg.