Let $A$ be a commutative unital ring. I am confused about a subtelty in defining the structure sheaf on $\text{spec}A$. The most conventional way seems to be to define a sheaf on a base, and then show that such an object determines a sheaf on the space. But it seems like different texts go about this in a very different way, and since one way is (at least seemingly) more complicated, I am wondering if I am missing something.
More mordern texts, for example Ravi Vakil's excellent book here proceed as follows:
For $D(f)$ a distinguished open in $\text{spec}A$, define $\mathcal{O}_{\text{spec}A}(D(f))$ to be the localisation of the ring of functions on $\text{spec}A$ (implicitly defined to be the ring $A$) at the multiplicative set of functions which do not vanish outside of $V(f)$. One can then verify that there is a canonical isomorphism $$ \psi: A_{f} \longrightarrow \mathcal{O}_{\text{spec}A}(D(f)). $$ This is also the the case in some older texts such as Hartshorne.
On the other hand, in sources like the Stacks project and EGA, the author proceeds by defining the presheaf on a base by setting $\mathcal{O}_{\text{spec}A}(D(f)) = A_{f}$.
Of course the resulting sheaves are isomorphic.
But considering the first contains strictly more work and fuss, it makes me wonder if there is some genuine reason that makes it either necessary or cleaner. Why is it done that way?
I am wondering if it has something to do with how one glues affine schemes later on, since in the gluing process it is not enough for things simply to be isomorphic, the isomorphism itself matters since they must agree. It seems like the former is taking care of that in the structure sheaf, while the latter would involve being more careful when gluing. Is this correct, or have I completely missed somethign trivial?