ag.algebraic geometry - Is Hartshorne's definition of the category of varieties natural?

I would just comment, but I'm a new user so I can't.

I believe you are referring to Hartshorne's definition on p. 15:

"A variety over $k$ is any affine, quasi-affine, projective, or quasi-projective variety as defined above."

The reason he defines things this way is that in section I.1 he defined affine and quasi-affine varieties; in section I.2 he defined projective and quasi-projective varieties. You are right that he hasn't made very clear the relation between the two. Exercise I.2.9 on projective closures hints at the relation, but it hasn't been made completely precise (he just uses the terminology "identify"). To make this precise, we need to say what the morphisms between any two different varieties are.

Suffice it to say, the identification of Exercise I.2.9 given by projective closure is in fact an isomorphism in the sense of section I.3, so all varieties are isomorphic to quasi-projective varieties.

If this is your first encounter with algebraic geometry, however, I'm not sure I would recommend chapter I of Hartshorne very highly. I made the mistake of thoroughly studying chapter I myself, and I think there are far better ways to spend your time.