ct.category theory - How should I think about correspondences?

The convention that the word "function" refers to something that takes exactly one value on every element of its domain is only about a hundred years old. A correspondence is a formalization of the older notion of a none-one-or-many-valued function, such as the square root or the reciprocal, in the framework of this convention. It's also possible to work with multivalued functions outside of this framework--Riemann surfaces were invented before this convention was established, to better understand multivalued functions in complex analysis.

In your version 2, we literally regard a correspondence as a (single-valued) function that takes an element x in X to a set of values in Y, by intersecting the defining subset of X x Y with {x} x Y and projecting to Y. In the more general version 1, we allow the set of values to be parameterized by something that's not necessarily a subset of Y.

To compose two correspondences, we say that g(f(x)) is the image of the set of values of f on x under the multivalued function g. This agrees with the usual fiber-product definition of composition.

From this point of view I think that the ring-valued correspondences on a set X won't themselves form a ring, for instance because there won't be additive inverses. (E.g. the function whose set of values on each element is the whole ring has no negative.) In category theory language the point is that the cartesian product is just some monoidal structure on correspondences, not necessarily a categorical product, so Cor(X,K x K) is not the same as Cor(X,K) x Cor(X,K).