Is a video game story good because the events and characters that exist beyond player control are well-written? Is it good because of how well the story and characters adapt to player choice? Or is it good because the game gives the player the tools to craft their own smaller stories purely through experimentation in gameplay, apart from the overarching narrative?
Maybe all of them. Maybe none. Damned if I know all the answers.
One of my favorite small examples is from Brothers: A Tale of Two Sons. It's a pretty simple game that expresses its story through gameplay mechanics and controls in a way that caught me completely off guard. That is an example of a game telling its story in a way that absolutely would not work in any other medium.
Games are an extremely versatile medium. And I shouldn't still be awake, as I'm barely functioning, so I'll reply to this real quick before I really, officially go to bed like I said I would, but for me the most innovative game I have ever seen is The Last Express. Here is a game that literally runs on a real-world timer, meaning that unless you play hundreds of times, no playthrough will be the same. You have to think on your feet, you can follow specific storylines, you can listen in on conversations in foreign languages just like you were on a real train at the time, you can go and watch an entire concert if you want and nothing else, and so on and on and on. A game that truly functions in real time and makes proper use of that is one of the most innovative, ambitious ideas games have ever had. It is the truest evolution of the concept of the adventure game, and it completely tanked commercially. As a piece of art meant to take you through a specific place in a specific time period it is unrivaled.
And then there's Mother 3, a game where every mechanic, every piece of dialogue, every character, every place, every enemy, etc. etc. all exist simply as a showcase of one man's writing talent and style, a style and point of view that is wholly unique from anyone else. There is literally no truer piece of artistic expression in games. The game is literally as much of a personal vision as a game could possibly be. Shigesato Itoi is a jack of all trades, he is revered all over Japan, and he is a complete genius as a writer. That's not even going into the actual story and themes of the game itself, which are stunning and a world all their own.
I could go on and on about games like Grim Fandango which makes Pixar movies look like shit, or A Night in the Woods, or Ico, and so on and so on, and I would, but I'm falling asleep now.