For me it was in Chrono Trigger.
In the scene where Lucca´s father, Taban, explains why her mother can´t walk.

In the beggining of the game, at certain times Lucca says that her mother Lara don´t usually go out and I began wonder why she was always at the same spot, at the same chair all the time, inside the house.
Later, her father Taban talked about some sort of machine he was working on, and her wife wanted to dust it off. The machine accidentally ignated and pulled her dress, Lucca couldn´t stop the machine. Taban was so consumed by regret and remorse that he became to be a very different person.

After that scene, I just couldn´t stop thinking about the scene and how horrible it would be to a child seeing this kind of thing happening to her mother, unable to do anything about it. It was a moving story.

Ahh yes, the terrible dive off the cliff.  That combo was depressing.

I do remember turning off the SNES twice when getting to the World of Ruin, convinced I had done something wrong.  I thought maybe I had to lose the boss fight, so I went back and let them beat me, hoping it would trigger some event that stopped the destruction.  Then when I realized that didn't work, I played through only to lose the mini-game.  Turned it off as she jumped because I couldn't handle it.  Went back through the floating continent and finally won the mini-game the next time I turned it on.

I think the first time was FFX, some of its CGIs including the ending. Till that point the other games hadn't really done it for me, including "idgaf"ing over FF7. Pre-ps2 era I was "fun-only" i guess, or whatever you want to call it
True turning point though was the ending of ICO, and then the ending of SOTC once again.

The one that hit me the most would be Xenoblade Chronicles. The opening left me feeling sad and I felt just how futile trying to hold back the Mechon was. The fear was real!!!

Then there was the attack on Colony 9 incident with a certain character and the awakening of Shulk. The moment Shulk awakens his first skill. My emotions were stretched between sorrow for the loss of a character, and hope that the Homs may have a chance at survival.

Then fighting the Face mechon after the revelation of what they are was pretty heart wrenching, at least for Shulk and Sharla. Metal Face had it f'ing coming though! (Yes, even some of the main badguys had pretty good back stories.)

The Di....

The tele.....

the ending...

Well crap, there is so much of the game I would have to spoil, but the ENTIRE 100+ hours you spend with the game, you always have motivation to go on. You always want to see what happens next, and it's always a rewarding experience when things are resolved. Maybe not always in a good way, some resolutions are devastating, but it is definitely a roller coaster ride you will be glad you experienced.

It still actually brings a tear to my eye... but that wasn't the first time a game got me in the feels.

The first experience I had with real true emotion I think was Illusions of Time (gaia in the states) when you have to play russian roulette against an NPC to allow you to progress in the game, the person you are playing against is a bad guy.... and you have to beat him in a game by picking out drinks, one of which is poisoned..... but your character in the story has physic powers and knows which drink is the safe one, regardless of your choice to pick you will always get the safe glass.... but after doing it and leaving just one glass remaining with the poison the person you are playing against congratules you.... and raises the final glass to you, drinking it and collapsing in front of you even though you have won and he admitted defeat, it was just the fact that you killed another person in your actions but not with fighting... not swinging a sword but just by watching him drink a glass of poison that you cheated your way out of. Really got me, that game is fan fucking tastic btw if you guys ever go on a SNES Rpg binge I highly recommend it.

It starts at 1m 30s

I have to agree with you both. I really don't like FFVI that much but when the World of Ruin is created, I felt really bad for the characters. Some of them just stopped caring and others actually did care but you had to get it out of them. It really captures the human emotion really well. And that's what ultimately creates emotional moments in gaming.

Spoilers but uh Final Fantasy IV
Two moments, the first is when Rydia is still a kid and Cecil and Kain unwillingly burn her whole town and she's crying over her dead mother, that moment when she is running away from them and yells "leave me alone, I hate you"
That was like omg like you could just feel how angry she was

And then the other part is when Cecil and everyone is getting absolutely hammered by Golbez and they're all about to die but then all of a sudden he gets hit by a mist dragon and Cecil gets healed and Rydia comes back and joins the party and says "don't worry, you'll be fine" as an adult (she gets thrown into the ocean pretty early on in the game and then comes back at the best part)

^This was such an underrated game. The Tellah casting Meteor scene being the first actual non reversable death in Final Fantasy  or the Porom, Palom sacrifice. 

FF7 - Aerith

FFIV has many "deaths", Tellah, Palom, Parom, Cid, Yang, Rydia. In the end, everyone pulls through except for poor Tellah who remains dead.