The bracket you linked is the way a lot of people believe double elimination is supposed to be played, but it makes little sense. In that format, the winner of the winner's bracket would play TWO fewer games, and then STILL be in a win-one-out-of-two format for the championship. That's a ridiculous advantage, and its why I don't even like 4-team double elimination.
You want something really confusing? You should look up the brackets for the old 6 team double elimination regionals. In that format, you have a 1-0 team playing an 0-1 team in the SECOND round. You can wind up with one 2-0 team, and four 1-1 teams. Quite confusing.
I played a lot of Dixie league baseball as a youth, and all of our 8 team tournaments followed the old CWS format. It was a lot of fun for the players, too.
OTOH, I once played in a corporate volleyball tournament that was 8 team double-elimination. We had four courts so everyone could play at once. My team was 2-0 and lost the winner's bracket final, and then I found out they intended to run the tournament they way you posted, so all the 1 loss teams had to eliminate each other while the 3-0 team sat around. I protested and whipped out the CWS format. Everyone was starting to agree with me, primarily because they realized it would save time. Then the park manager told us he needed our courts and we were limited to one court for the duration of the tournament. The 3-0 team then insisted we stick with the old format, and my team did not have enough energy to win two straight at the end.