That's literally moot
I don't see how. Under your definition, it would seem that there is not such a thing as interactive fiction.
The story teller exercises fiat over such improvisation. In a game, the rules of the game hold fiat.
No, they don't. The rules of the game are actually only a small part of any RPG. Equally important to how the game is actually played is the social contract, the rulings of the games moderator, how the game was prepared for, what the expectations of the players are, and of course the fact that in most RPGs the moderator is specifically given permission to not only improvise rules but overturn them at need. This is why two groups playing exactly the same rules system can be playing completely different games and have completely different game experiences, to say nothing of obviously experiencing and producing completely different stories.
A game is an objective reality, not a subjective one, at least to the extent the mechanics are involved.
What does that even mean, and how is it even relevant? A book and paper or a film and a screen are objective reality, but they are mediums for telling the story. They have mechanics and authors use them to create stories. The only real difference is that the game medium produces collaborative stories.
For me a game is a machine for telling stories - the story teller - not the story.
That's splitting the hair so fine that I don't really see how it makes a difference. If the game is a machine for telling stories, the collective framework by which the collaborative story teller's abide, then the process of playing the game is still itself a story and inseparable from experiencing a story. It would be like you arguing that a novel is not a story, because it's book and a book is a thing with paper and words on it.
Once it is received as a story, that no longer happens. Whereas a game can always be different on every occasion.
I have no idea what you are trying to say, unless that it is the game doesn't become a story unless you play it. But it doesn't by the exact same measure become a game until you play it either. Before that it is just a rulebook and some dice and other props of the game.
You keep going for a while after that, splitting the hair in even finer ways, but I think you could simplify things greatly by just saying that D&D is a game and a story. As soon as you actually play the game, it becomes both, and the story and the game interact with each other in feedback loops that are inseparable.