Players narrate what they do, making choices that alter the direction of the story, sometimes in completely unexpected ways. Dice rolls similarly add new elements to the story, so that both they and I have to improvise. At the end, a story has definitely happened, and we usually have a great time.
All activities produce stories, that is just a natural byproduct of events happening in a sequential order.
Yesterday, I played Team Fortress, I flirted with this girl playing Soldier, and then I popped kritz on her and she bathed me in the blood of the enemy team. Story! Or, in-universe: yesterday, a group of 12 mercenaries hired by Builders League United took control of a gravel processing plant in the New Mexico desert with minimal casualties, soldier Jane Doe (callsign "♡Diana♡") and known war criminal Fritz Ludwig (callsign "Alice can't hit pipes") earned commendation medals from the corporation. Story!
OK, I know I'm being cheeky. But in, say, Warhammer, people been writing, uhm, literary (?) battle reports where unit movements and shootouts are woven into narrative since before my ma even decided she wanted a kid, but calling Warhammer a storytelling game would be lunacy.
D&D isn't a storytelling game because it provides no tools and no incentive to engage with Shared Imaginary Space through the lens of storytelling, and the rules themselves don't act as such a lens either.
What would be an example of a game that is a storytelling game?
The best storytelling game I know of is probably Daniil Shipaev's
MUJIK IS DEAD. It directly provides prompts for situations (like "Falling in love" or "Money problems"), and the players purposefully weave a tragic story of suffering and misery. One player takes on the role of the titular Mujik (translated: rough, tough, "real" man), the others are playing as his destructive drives. At the start of each scene, a random prompt is chosen and Mujik describes the situation, the other players tell what they think should happen next, they all roll dice and the winner narrates the outcome.
The story isn't just a byproduct of the gameplay, it
is the gameplay. The dice are rolled not to determine success or failure, but to decide who has narrative authority.
A more mainstream example would be Fate, where
- The players can directly declare facts using Fate points (like "I'm a *Seasoned Explorer, so of course I find a secret passage here, hidden behind a loose brick wall") and gain Fate points if the fact is detrimental (like "I'm a *Wanted Fugitive, so of course our contact will recognize my face").
- Storytelling concepts, like genre, tone, tropes can be codified directly into aspects, and leveraged in the same way. Your game being an *Amateur neo-noir webseries, or your character being a *Designated Love Interest are all part of the mechanics, and players can and should leverage them.
- Most importantly, players have an option to back out of pretty much any situation, so they can safely raise stakes and create trouble.
An example of a game where rules act as a storytelling lens would be Horror Movie World (or Dread that you've mentioned upthread): the rules are specifically designed to replicate slasher flicks, where a colorful cast gets picked off one by one, being competent is the easiest way to die, all that.