The definition in Hartshorne isactually , surprisingly, the most natural one.
Let's approach this in a following way. If you think meditate about what Grothendickian algebraic geometry is, you grok suddenly that there are it tells a story of some forms that are desribed algebraically but imagined geometrically. The first such form is A^1, affine line. You should Take a time to focus on both the techniques (meditate on
k[x]) and content (a geometric qi, if you wish)wish) related to affine line.
Now let's go to the form called
P^1. You should be able to imagine
P^1 — it's simply a sphere, or a space of lines on a plane or whatever image you like. Now, what's the technique that would describe this formally? If you think about it, you'll come up with the same idea — it's something made out of two simpler things, the
Ok, now lets think about maps between these forms. E.g. take arbitrary point-to-point mapping. It's clearly something wrong, not beautiful, because I've thrown away all the intuition I've asked you to create. So, let's concentrate better. Our structure is algebraic. For an affine manifold, this has been formalized as "polynomials go to polynominals". For a smooth manifold it's been formalized as "locally, smooth functions go to smooth functions".
Now, if we combine so, we get the definition of "locally, polynomials go to polynomials" — and this is the definition that is useful in practice. The Hartshorne simply has it written in an abstract language that can be applied to any scheme. That who masters this example shall have no fear.