Since my quote began this thread, I thought I should chime in.
In my book I discuss four approaches to storytelling in games from the most restrictive to the least. The first is Traditional (Linear); the second is Branching (Linear-Thinking); the third is Web (Simple Non-Linear); the last is Modular (Non-Linear) the approach I've been using whenever possible since 1995. It's what I've been teaching at GDC tutorials, conference lectures and now at Indiana University. I say "whenever possible" because I'm a professional game designer and writer, and often I'm expected to use--or am forced to use due to the linear material from another medium I may be adapting--the less satisfying approaches.
Modular storytelling and systemic (as opposed to tightly-scripted) game writing fit the way most players want to play games. Players want to feel their choices have meaning; that the game world is changed by those choices. Modular storytelling allows this true interaction between player and narrative in a way linear and branching never can.