In pg. 60 of Categories for the Working Mathematician:
Let $D$ have small hom-sets. A representation of a functor $K: D \rightarrow \mathbf{Set}$ is a pair $\langle r, \psi \rangle$, with $r$ an object of $D$ and
$$ \psi : D(r, -) \cong K $$
a natural isomorphism. The object $r$ is called the representing object. The functor $K$ is said to be representable when such a representation exists.
Question: In this definition, does the notation $\psi : D(r, -)$ just mean that $\psi$ is another name for the functor $D(r, -)$?
No. In fact you should read it as $$\psi : D(r,\_)\Rightarrow K$$ so that $\psi$ is a natural transformation. Mac Lane uses $\psi : D(r,\_)\cong K$ instead to indicate that $\psi$ is an isomorphism between the two functors (which he actually repeats right after the equation anyway).