Why are these two definitions of PPAD equivalent?
For issues with the cited paper, (and hence this answer) see Does PPAD really capture the notion of finding another unbalanced vertex?

Yes. These two problems are equivalent and hence PPAD-complete. The reduction is given on page 505 of Papadimitriou's original 1994 paper (pdf) introducing End of the Line. This is valid even if the degree of the graph is exponential, provided we are given an "edge-recognition algorithm" and a "pairing function". This is also mentioned on the same page. The reduction is given for PPA (the undirected version of PPAD). It can be extended to PPAD as well.