The "modern definition" you are referring to is indeed classical and predates von Neumann's innovation. The equivalence approach is implicit in the work of Cantor, but it was made explicit by Russell (and is based on Frege's similar treatment of cardinals as equivalence classes).
Chris Eagle pointed out in his comment that with the equivalence relation definition of ordinals, no ordinal is a set except 0. This is indeed the case in Zeremlo-style set theories. But in set theories such as Quine's $NF$ or Quine-Jensen's $NFU$ [in which there is a universal set], the equivalence approach leads to sets. The consistency of $NF$ relative to a $ZF$-style set theory is still open, but Jensen showed the consistency of $NFU$ relative to a small fragment of $ZF$ in 1969, see here for more detail.
P.S. $NFU$'s ability to handle "large objects" makes it an interesting vehicle for the foundations of category theory. Solomon Feferman has done significant work in this area, if interested, see his paper, and talk.
