The latter. Double elim grand finals go like this:
If the winner of the winner's bracket wins (lololol wordplay) the first Best-of-3 match, he wins the whole tournament (his opponent has now lost two matches and is eliminated).
If the winner of the winner's bracket loses the first Best-of-3 match, that is his first tourney match loss, so he is not eliminated and gets another chance. They then play a second Best-of-3 to decide the grand champion.
So in the case of IpKai's tourney, Barton lost the first match 0-2, but came back and won the second match 2-0, thus winning the tournament.
