I understand the definition of a minority pawn attack, is when one side has less pawns than other in a flank of the board (or any sector) and advances them to trade them with the oponent. The question is why would I want to do that? I imagine there is a different purpose of that for every game, but possibly there is a general reason that applies to various games. I heard here: https://youtu.be/ziCLlmz0ATo?t=4m26s that minority attack serve the purpose of leaving pawn weakness. Probably leave an isolated pawn?
The point of the minority attack, you have correctly pointed out, is to force pawn weaknesses. Usually, the weakness that you are going for is to create a backwards pawn. This weakness is very difficult to deal with for your opponent. Normal play is to gang up on the pawn, double your rooks along the file.
A minority attack begins when you have fewer pawns on one side of the board (usually the queen side) and one more pawn on the other side. When pawns are "unbalanced," the game often comes to a "race" between the players' respective strengths on the king and queen sides. If you can tie up the opponent's supposedly "strong" side with a "minority" attack, the strength on your strong side should decide the game in your favor.
So the essential goal of a minority attack is to create a situation where your one fewer pawns is worth as much as your opponent's pawns that number one more (that is two is worth three). This is done by creating a weak enemy pawn that is subject to heavy attack, or is captured outright.
In "Chess Fundamentals," Capablanca expressed it another way: "A unit that holds two." Say you are Black with a pawn on a7. White has two (queen side) pawns on a2 and b2, He unwisely moves a4 (he should have moved the "extra" b pawn first). You play a5!, and now your one pawn is worth two of White's. His b pawn is "backward," and if he moves it to b4, you play a5Xb4, and your new "b" pawn is one step ahead in the queening race against the pawn on a4. So White's two queen side pawns are effective "tied up," and Black's extra king side pawn should win for him.