Seven. As usual, the alleged Wikia has missed its mark.
Philosopher's Stone (2)
Harry was speeding toward the ground when the crowd saw him clap his hand to his mouth as though he was about to be sick — he hit the field on all fours — coughed — and something gold fell into his hand.
"I've got the Snitch!" he shouted, waving it above his head, and the game ended in complete confusion.
Harry Potter and the Philosopher's Stone Chapter 11: "Quidditch"
Up in the air, Snape turned on his broomstick just in time to see something scarlet shoot past him, missing him by inches — the next second, Harry had pulled out of the dive, his arm raised in triumph, the Snitch clasped in his hand.
Harry Potter and the Philosopher's Stone Chapter 13: "Nicholas Flamel"
Harry is unconscious for the last match, so does not play.
Chamber of Secrets (1)
Harry took his remaining hand off his broom and made a wild snatch; he felt his fingers close on the cold Snitch but was now only gripping the broom with his legs, and there was a yell from the crowd below as he headed straight for the ground, trying hard not to pass out.
With a splattering thud he hit the mud and rolled off his broom. His arm was hanging at a very strange angle; riddled with pain, he heard, as though from a distance, a good deal of whistling and shouting. He focused on the Snitch clutched in his good hand.
Harry Potter and the Chamber of Secrets Chapter 10: "The Rogue Bludger"
Because of the Quidditch Cup being cancelled, on account of all the attempted murder, this is the only game Harry plays.
Prisoner of Azkaban (2)
Something silver-white, something enormous, erupted from the end of his wand. He knew it had shot directly at the Dementors but didn't pause to watch; his mind still miraculously clear, he looked ahead — he was nearly there. He stretched out the hand still grasping his wand and just managed to close his fingers over the small, struggling Snitch.
Harry Potter and the Prisoner of Azkaban Chapter 13: "Gryffindor versus Ravenclaw"
Harry threw himself forward, took both hands off his broom. He knocked Malfoy's arm out of the way and —
He pulled out of his dive, his hand in the air, and the stadium exploded. Harry soared above the crowd, an odd ringing in his ears. The tiny golden ball was held tight in his fist, beating its wings hopelessly against his fingers.
Harry Potter and the Prisoner of Azkaban Chapter 15: "The Quidditch Final"
The only other game Harry played was against Hufflepuff, but he was rendered unconscious by a Dementor and failed to catch the Snitch.
Goblet of Fire (0)
No quidditch this year; move along.
Order of the Phoenix (1)
"It was that thug Crabbe," said Angelina angrily, "he whacked the Bludger at you the moment he saw you'd got the Snitch - but we won, Harry, we won!"
Harry heard a snort from behind him and turned around, still holding the Snitch tightly in his hand: Draco Malfoy had landed close by. White-faced with fury, he was still managing to sneer.
Harry Potter and the Order of the Phoenix Chapter 19: "The Lion and the Serpent"
Harry gets banned from playing Quidditch after this game, so no more in this book.
Half-Blood Prince (1)
"YES!" Hairy 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.
Harry Potter and the Half-Blood Prince Chapter 14: "Felix Felicis"
Harry plays one more match this year, against Hufflepuff. However, he is rendered unconscious by a bludger (aimed by his team-mate McLaggen), and fails to catch the Snith.
Harry was in detention with Snape during the last match, and so did not play.
Deathly Hallows (0)
No Quidditch was played in this book.