It's hard to say but I'm going to go with Ocarina of Time. I had played games up to that point but it had always been in an extremely casual manner. I never played on my own time (outside of Pokemon which I was obsessed with. When I did game, it was almost exclusive to when hanging out with friends with the rare exception being playing handheld games on the school bus or on long car rides. Ocarina of Time changed all that. I became completely enraptured by everything in the game and Zelda became a huge obsession of mine for a good chunk of my late childhood (fueled by a mixture of my love for OOT, WW, MM and ALTTP, my hype for Twilight Princess, and their relation to both my friendships and my creative endeavors / imagination (writing stories, drawing, etc)). However, I'm hesitant to say it's what made me a gamer as my tastes were largely just Zelda, Pokemon (for a short period of time predating my infatuation with Ocarina) and any Nintendo title that referenced Zelda in some way (I got a Gamecube with Melee just to play as Link and Sheik and picked up Animal Crossing the following year just because of a few Zelda references).
However, when it came to expanding my tastes, I'd credit a slew of 3rd party titles that hit the Gamecube in 2003. The first of said titles was Soul Calibur II, which I admittedly bought day one because of the Link cameo, but would become a multiplayer staple alongside Super Smash Bros. Melee. As the year went on, titles like Viewtiful Joe, Beyond Good and Evil and Prince of Persia: The Sands of Time would all broaden by horizons beyond just Zelda. In fact, the following year I would get my first non-Nintendo console (the O.G. Xbox) and it was from there that I would gain my true love of gaming. I gamed most heavily in my early teens while falling out of it in my mid to late teens. I would still play games on occasion but I wouldn't spend anywhere near as much time on them and the amount of games I played to year usually peaked at about 2 big titles a year. I though for the longest time that I was almost done with video games until 2011 when I picked up LA Noire. It completely revitalized my love of games - so that too could be seen as a title which "made me a gamer". Again, it's much to hard to pinpoint just one title...
tldr;
Up to age 7: Casual interest in video games but nothing to serious or commital
1998-2000 (7-9): Almost exclusively played Pokemon games
2000-2003 (9-12): Started to play a wider variety of games but was mostly just obsessed with Zelda (Beginning with my discovery of Ocarina of Time and peaking with the release of Wind Waker).
2003 (12): Started to gain a love and appreciation for a wide variety of games, started playing games not developed by Nintendo. At age 13, I got my first non-Nintendo console.
2004-2005 (13-14): Years of my life when I was most immersed in gaming and gaming culture; played a wide variety of games and extremly knowledgeable on the topic
2006-2010 (15-19): Fell out of love with games, played on occasion but could hardly identify as a "gamer"
2011 (20): Picked up LA Noire; revitalized my love of games through its novel gameplay, moody atmosphere, engaging story and cinematic style.
However, when it came to expanding my tastes, I'd credit a slew of 3rd party titles that hit the Gamecube in 2003. The first of said titles was Soul Calibur II, which I admittedly bought day one because of the Link cameo, but would become a multiplayer staple alongside Super Smash Bros. Melee. As the year went on, titles like Viewtiful Joe, Beyond Good and Evil and Prince of Persia: The Sands of Time would all broaden by horizons beyond just Zelda. In fact, the following year I would get my first non-Nintendo console (the O.G. Xbox) and it was from there that I would gain my true love of gaming. I gamed most heavily in my early teens while falling out of it in my mid to late teens. I would still play games on occasion but I wouldn't spend anywhere near as much time on them and the amount of games I played to year usually peaked at about 2 big titles a year. I though for the longest time that I was almost done with video games until 2011 when I picked up LA Noire. It completely revitalized my love of games - so that too could be seen as a title which "made me a gamer". Again, it's much to hard to pinpoint just one title...
tldr;
Up to age 7: Casual interest in video games but nothing to serious or commital
1998-2000 (7-9): Almost exclusively played Pokemon games
2000-2003 (9-12): Started to play a wider variety of games but was mostly just obsessed with Zelda (Beginning with my discovery of Ocarina of Time and peaking with the release of Wind Waker).
2003 (12): Started to gain a love and appreciation for a wide variety of games, started playing games not developed by Nintendo. At age 13, I got my first non-Nintendo console.
2004-2005 (13-14): Years of my life when I was most immersed in gaming and gaming culture; played a wide variety of games and extremly knowledgeable on the topic
2006-2010 (15-19): Fell out of love with games, played on occasion but could hardly identify as a "gamer"
2011 (20): Picked up LA Noire; revitalized my love of games through its novel gameplay, moody atmosphere, engaging story and cinematic style.