I think the term “goal and approach” as a description of the set of techniques I, Ovinomancer, and others employ is causing undue miscommunication. And I accept a large part of the responsibility for coining it. Let me try to clear some things up.
Those of us who use these techniques strive to at all times follow the flow of play described in the beginning part of the PHB, wherein the DM first describes the environment, then the players describe what they want to do, then the DM determines the results (possibly calling for a die roll to help in this determination), and then describes the results. I don’t think this particular part of what we do is especially controversial. I think pretty much everyone who runs 5e does this, to a certain extent. There are two major places where I think the two sides differ: how we interpret “a player describes what they want to do” and the methods we use for determining the results of said description. I think where things are getting muddled is in the conflation of these two points.
When those of us on my side of this debate read “the player describes what they want to do,” our interpretation is that this need be a description of the character’s activity in the fiction. “I cast fireball” is a description of the character’s activity in the fiction; the fireball spell exists in the world of the fiction, and casting it is understood to be an activity that involves particular magic words, gestures, and uses of particular materials to produce a particular result. “I make an Insight check” is not a description of the character’s activity in the fiction.
Related to, but separate from this, we strive to eliminate the need for the DM to make assumptions about “what [the player] wants to do” in order to determine the result. In the case of spellcasting, this generally requires very little effort. The rules provide explicit instructions for what mechanics to employ to to resolve the particular effects produced by the particular activities understood to be performed by a character casting the spell. In the case of an attempt to recognize that an NPC is lying, it is more difficult to resolve the outcome without making assumptions about the character’s activity in the fiction. To do so, the DM needs to know specifically what activity the character is performing in the fiction. So, where specific resolution procedures are not provided by the rules, and the DM is expected to use their best judgment to decide which mechanics, if any, to employ, we ask that the players communicate both what they as a player want to achieve, and what their character is doing that they hope will result in the desired outcome, in order to make our determination of the results as easy and assumption-free as possible.