I discovered that $$R \textrm{ is an equivalence relation on } A \;\equiv\; \langle \forall a,b \in A :: aRb \:\equiv\: \langle \forall x \in A :: aRx \equiv bRx\rangle \rangle$$
The nice thing about this alternative definition is that it says, "equivalence of $a$ and $b$ means that it doesn't matter which of the two you substitute", which seems to be the essence of equivalence (cf. Leibniz' law).
Is this alternative definition well-known? useful? used? Update: Or does anyone perhaps have a literature reference?
(I originally asked this same question here: Is this alternative definition of 'equivalence relation' correct? But that question turned into a correctness discussion. Therefore I have posted this as a new question.)
This equivalence was known at least by 1991, although likely beforehand:http://www.mathmeth.com/tom/files/equivalence.pdf
The author of this note calls it a 'beautiful characterization', and I agree with both of you that it's very cool!