From the explanation on flesh memories from Rufus Scrimgeour, it seems as if the first seeker to touch the snitch wins the game.
"Correct," said Scrimgeour. "A Snitch is not touched by bare skin before it is released, not even by the maker, who wears gloves. It carries an enchantment by which it can identify the first human to lay hands upon it, in case of a disputed capture. This Snitch", he held up the tiny golden ball, "will remember your touch, Potter.
So the first seeker to "lay hands" on the snitch wins, right?
There is a contradiction though. When Wood explains the rules of the game to Harry, he specifically says a few times that in order to win the game the seeker must catch the snitch.
"This," said Wood, "is the Golden Snitch, and it's the most important ball of the lot. It's very hard to catch because it's so fast and difficult to see. It's the Seeker's job to catch it. You've got to weave in and out of the Chasers, Beaters, Bludgers, and Quaffle to get it before the other team's Seeker, because whichever Seeker catches the Snitch wins his team an extra hundred and fifty points, so they nearly always win. That's why Seekers get fouled so much. A game of Quidditch only ends when the Snitch is caught, so it can go on for ages...
Further more, there is a game that happens where the Slytherin seeker touched the snitch first (but didn't catch it), and the Gryffindor seeker (Harry) caught the snitch first, and Gryffindor won.
"Oi, Harper!" yelled Harry in desperation. "How much did Malfoy pay you to come on instead of him?"
He did not know what made him say it, but Harper did a double-take; he fumbled the Snitch, let it slip through his fingers, and shot right past it. Harry made a great swipe for the tiny, fluttering ball and caught it.
"YES!" Harry yelled: wheeling around, he hurtled back toward the ground, the Snitch held high in his hand. As the crowd realized what had happened, a great shout went up that almost drowned the sound of the whistle that signaled the end of the game.
Either the first seeker to touch it wins, and when Harper fumbled the snitch Harry wasn't supposed to win but he won anyway for some reason, and when Wood was explaining the rules to Harry he got confused and he taught Harry that he needs to catch the snitch.
Or the first seeker to catch it wins, and the flesh memories charm is made in a way that it detects only the first person that catches the snitch and not necessarily touches, and Scrimgeour was getting confused between touching the snitch and catching it.