I think the example you are looking for is the game of DeusEx. The player does actually chance the flaw of the story by acting one or another way, you can change the future and the story that you will live by the style you play, you can even pass the whole game without even shooting just one time. You have the freenes of doing whatever you want to, and still can finish the story, just in one bad or a good way, which will in the end change the curse of the story.

Anyone who played Deus ex(at least 4 times) knows what i am talking of. This game is yet unique, even if there are other games(e.g. Farenheit) out there to claim the story changes by your doings, it does not really. Just the end changes, infact in deus ex you can play the whole game in many different manners, and even suddenly betray a group you have worked for.

All this coombined with the well thought and deep story, it is just the perfect game, till today. No other game(not even Gothic 1+2) could reach DeusEx-Level.