I know TLOU is the flavor of the month now, but I didn't expect so many to just focus on that game and ignore all the great games even in just the PS2 generation....
Silent HIll Series: So much of the story was told without reliance on cutscenes. Those scenes played an important part in the story, but memos/notes, locations, and imagery told volumes. It is really the Blade Runner of videogames ... showing us so much but rarely if ever explictely telling us what is going on. It is the job of the viewer/gamer to interpret the story based on visual clues.
Shadow of the Colossus: Sames as Silent HIll, it showed us so much without forcing cutscenes down our throats to explain everything to us.
Deus Ex series: So much to learn from notes/ect. and supporting characters about the world of Deus Ex.
There are others, but those are some of the best examples I can think of moving/powerful stories that did not rely on cinematic cutscenes/presentations.
