Mass Effect for example is a game that obviously relies on the storyline, to me, everything past the writing was extremely bland in that game, and I still love the series.
Some games make you feel like you're in an actual battlefield slash wasteland slash setting by storming you with realism, while some other games make you feel enrolled with the character, therefor, having you feel like it's your duty to save the galaxy or defeat the daily baddie.
My introduction to gaming was in the "Here's what's going on now go out there and kick some ass" type of videogame, when good storyline was a very casual one night stand kind of thing, and I still value games that leave it up to you whether to explore the storyline too much or not.
As much as I personally love the Silent Hill series, I always hated the over-analytical plot, which always had many different approaches, most of which led your understanding of the storyline to a dead end and a million comprehensive threads to try to get a recap of exactly what's been going on through the last 4 games, otherwise, you won't "fully appreciate the game", but in the end, I came to the conclusion writers just decided to give you the facts and make you feel like there's a lot of depth, when you're actually having a straight forward line of events.