I define the two approaches as pattern finding and story telling.
In the first, the Ref is relaying a pattern behind a screen. It is a reality puzzle game with a cooperative simulation game behind a screen treated as a code. Irrelevant answers are responded to with a "yes", but cannot than be contradicted. This contradiction can even be put to the players, if they choose not to see a contradiction. Ref: "You said your dance engaged your legs, feet, arms and torso. Are you doing something different now?" Player: "Yeah, I want to be able to shoot my bow, so I'm just kickin' out the steps now".
The second example is pure storytelling, not code breaking. There are no consequences except as each player chooses to see stuff as consequential. They are each telling a story with the pragmatic understanding each has something to do with the other. There is no authority to do so, but what each player chooses authority to mean.
I use the Tic-Tac-Toe example to explain this a little more clearly. The first opinion, a critical realist pragmatic perspective, is playing Tic-Tac-Toe and trying to find a way to achieve the objective of three X's or three O's in a row. The second opinion, a naive realist perspective, is seeing the underlying pattern in the game. Unless one makes an error, they should be able to win, if not tie every game they play. The third opinion is a postmodern irrealist. There is no ability to recognize patterns as patterns in the brain do not exist, so each person is simply expressing their selves with X's or O's. At a certain point, they may decide whose story is "better" and that person "wins". So all games and contests are popularity contests, not reaching a predetermined goal (the last is unknowable).
D&D started as a hidden simulation game behind a screen. It's sort of like the game Mastermind where one person relays the four pegs behind the screen. In the modernist sense, objectively relaying color and spatial positioning is not wavering from the pattern within the scope defined. The predetermination of that code is subjective, however. In the postmodernist sense, objective reality is an illusion and should be engaged with ironically or inauthentically. From this point of view the person behind the screen is an oppressor and limiter of others' freedom to express their story. To be a DM is to be a tyrant.
Currently, the postmodern opinion is being treated as an absolutism. Though philosophers like Rorty have said there really is no difference between the two except as yet another perspective, it often falls into atheistic fantasies towards truth rather than agnostic. I think of it like a glass half full versus a glass half empty choice.
As in the OP's examples, if one plays a role playing game as a simulated reality behind a screen, they are treating it like almost all people treat our own reality - in a pattern finding manner. This is like almost everyone treats computer output including computer game, except RPGs are larger in scope as non-programmed actions are programmed with each player's attempts, which are irrelevant to the hidden code. As a game of semantics, D&D isn't solving world problems, but it does push people in pragmatically beneficial behaviors. And as a language game it has the side effect of increasing participants' vocabularies.
Story games are from the "narrativist" point of view. There is a strong, absolutist belief running under the current game design theories of all games as stories, but it is not held by all. The "gamist" or ludological point of view is probably best found in those using math's game theory for game design, mathematics being the language of patterns. As story telling endeavors with devout attempts by players for apatternicity towards any underlying patterns in games like Tic-Tac-Toe, Chess, and Go, these games are likely the most unsatisfactory game designs. Telling stories on a grid-board with carved pieces or stones falls into what is often called austerity story telling. I can only imagine such play is abysmal for them. For those who are capable of recognizing not the whole, but the part of the underlying patterns in those games, they can be life long endeavors to increase mastery.
After all, I've seen DM's get very intense when the player starts adding in details without asking. OTOH, I've seen players fumble and stumble around because they're looking to the DM to provide more details instead of coming up with their own.
In a pattern finding game details are learned through attempts taken each turn as in Mastermind. In storytelling games the details are accepted or disallowed as each person chooses. If everyone at the table doesn't understand what type of game they are playing, then... Maybe the DM get upset. Or maybe the players don't realize they need to be the initiators. That goes for increasing the explored details as well as driving the action.