Mine is "Makenai Ai ga Kitto aru" by Yukie Nakama
It was used as the opening theme for Mega Man X4 in Japan.
I like the music used for the overseas opening a lot as well, but I think this was a good fit for the game.
What I really find surprising is I like the non-edited rendition; usually I get use to edited versions used.
Still, I like that this is my favorite because Mega Man X4 is one of my favorite games.