Question on equivalence of definitions of quasi-affine varieties

Question

I'm hoping to introduce some basic concepts of classical algebraic geometry to some students in a seminar soon. To avoid the usual ambiguity and confusion around the terms in this picture, I was hoping to present things in a more consistent, if atypical, fashion. Since I have a tendency to overlook some subtleties, I'd like to know if there are any issues with the following outline:

1. Fix a field $k=\overline{k}$ and define $\mathbb{A}^n$
2. Define the Zariski topology on $\mathbb{A}^n$ and use it to distinguish closed and irreducible closed subsets.
3. Define a morphism $f:X \to Y$ between subsets $X \subseteq \mathbb{A}^n$ and $Y \subseteq \mathbb{A}^m$ to be a set-theoretic map $f =(f_1/g_1, \ldots, f_m/g_m)$, where $f_i,g_i \in k[x_1,\ldots,x_n]$ and $\forall x \in X, g_i \neq 0$ (a regular map). Isomorphisms are biregular bijections.
4. Give the following extrinsic definitions: $X \subseteq \mathbb{A}^n$ is a quasi-affine variety if it's isomorphic to a closed subset of some $\mathbb{A}^m$, and an affine variety if it's isomorphic to an irreducible closed subset of some $\mathbb{A}^m$.

I think this captures the picture with some consistency, but I'm not sure if the definition of quasi-affine above is equivalent to the usual definition as locally closed subset. If $X$ is locally closed then I believe it has a closed image in some $\mathbb{A}^m$ (if it's an open subset of a closed set $X \subset \mathbb{A}^n$, then it can be written as a finite intersection of closed sets in $\mathbb{A}^{n+1}$). I'm not sure about the converse, but the idea that quasi-affine means we have a solution of a polynomial system minus some other solution set should mean it's a union of solutions somewhere.

Answer 1

$X \subseteq \mathbb{A}^n$ is a quasi-affine variety if it's isomorphic to a closed subset of some $\mathbb{A}^m$

This is not equivalent to the usual definition of quasi-affine; for example, $\mathbb{A}^2 \setminus \{ 0 \}$ is quasi-affine in the usual sense but not in this sense. (Personally I would call this "affine.")