Do the American & National leagues in baseball have different tie breaking rules?

No. Same rules.

However, MLB does treat standings ties differently depending on whether both teams are advancing to the postseason regardless (one as division champion, the other as the league wildcard) -- such ties are broken on paper, as would have happened with the 2009 NL West -- or when two teams are vying for only the one postseason berth, as with the 2009 AL Central.

When two teams are in line for one golden ticket, then they settle the matter with bats and gloves on the diamond, as God and Alexander Cartwright intended.

And that's all. Both situations have occurred before.

Same rules in the leagues. The reason you perceive a difference is that you're looking at two different situations. Dodgers/Rockies are both in the playoffs even if tied so there is no playoff. Twins/Tigers is different. Only one makes it. The other goes home. So there's a playoff.

No they do not. In the case of the Dodgers and Rockies, both teams were already assured of making the playoffs. In the case of the Twins and the Tigers, both teams are competing for just one spot, and thus tiebreakers determine where the play-in game is played. The loser of the play-in misses the playoffs.

There are different rules for these circumstances:

1. 3 teams for 2 spots (one is a division title and the other is the wild card). Here, the division title must be determined first.

2. 4 teams for 3 spots (two are division titles and 1 is a wild card). Again, division title first.

3. 4 teams for 2 spots (one division title and one wild card): Title is determined first, tiebreakers determine order of other play-ins.

Maybe they meant the Dodgers would have the tie-breaker at home.

That's because they were both in the playoffs no matter who ended up winning the division.

no its the same the twins have won more against the Tigers that is why it's there