In Natural Operations in Differential Geometry by Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$.
Is this is a precise definition, or just a hand-wavy characterization?
I would have expected that a differential operator would be required to be a sheaf homomorphism between the sheaves of smooth sections of $Y \to M$ and $Y' \to M'$. Is there any reason why KMS do not require a differential operator to be a sheaf homomorphism, but (apparently) just a function between the sets of global sections?
For the sake of people ending up here from search engines, here is a sketch of an answer.
The two definitions should be equivalent, at least when the operator on global sections is nice enough. To give a map of sheaves of sections, one needs to specify maps of stalks at every point in a compatible way. Using a partition of unity, a germ of a smooth section may be extended to a global section. Apply the operator on global sections, then restrict to the stalk of the same point. This turns out to be well-defined, proved by induction on the order of the differential operator (maybe the map of global sections should be continuous in Frechet topology for this argument, I don't know).
The order 0 case, i.e. linear operators, is called Serre-Swan theorem. The trick is to use linearity (or the Leibniz rule for higher order operators) for the multiplication by a bump function.