I loved playing video games for as long as I can remember. It was usually Nintendo games, and occasionally kids PC games in the 90's (Pajama Sam, Putt Putt, Etc.).
If I was to get to the point where I came to be the definition of a gamer, which I consider someone who can appreciate the different aspects of a game, I can't pin it on one game, but some games over time made me start to appreciate it.
Final Fantasy 7 was the first game that really opened my eyes. It was the first game I played that was so large at the time. It made me appreciate just what video games could become, in size and just the variety it could have. I was only 7 years old at the time, so most other things went over my head at the time, and even then the appreciation was still very little.
The next game was Tales of Symphonia. I was much older and was able to fully appreciate the world it made, as well as the characters and story. It stuck with me a lot and when playing other games after that, I started looking more into characters and the story.
The last game that really opened things up for me was Persona 4, which I played and beat my second year of college. It really showed how everything from characters, to story, to music, and everything can come together to make things very impactful. I play through it again every once in a while to find new things to appreciate. With this appreciation, I was able to go back to playing Super Metroid and actually comprehend why it is still great and what makes it great.
I still have a ways to go in being able to look into things even more deeply. From various youtubers to posters on this forum, I'm still very much a novice at looking at everything, and even that's giving me to much credit.
So long story short, Final Fantasy 7, Tales of Symphonia, and Persona 4 in that order made me a gamer.