Thus, there is a spectrum of impact a player can have on a game.
Games without narrative stories are the ones in which you can have the most impact: sports games, The Sims, and chess are examples. Playing with a big box of Lego™, with all the accessories, would be an extreme example. Your impact on the way the Lego is used and the possible outcomes is huge.
On the other extreme, you could have a game with two different endings. A decision you make near the end would determine the outcome. Here you'd have an impact, but on the impact scale, it would be much less than in The Sims. You could also have other forms of impact as to how you accomplish your missions in terms of stealth versus combat, the weapons you choose, the abilities you use, and even perhaps the geographic route you select.
One form of game isn't innately better or worse than the other; many players enjoy multiple forms of gameplay. And so one form of impact the player can have on the game isn't innately superior. The more of a "real" impact you have in the game, the less there can be anything that resembles narrative story. The more the game contains a narrative story, the more your impact in the game resides in the smaller details, but not the overall direction and shape of the outcome.