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.
(Emphasis added.) From my perspective the controversy on the bolded part stems from the insistence on striving to follow that flow of play "at all times", and whether or not that exclusivity is mandated/suggested/implied by the rules of 5e or is instead a style preference.
I don't think there is any way to objectively resolve that controversy, but I'm flagging it since you mentioned you didn't think the bolded part was controversial. (Admittedly, the degree to which such exclusivity is supported by the rules is a side issue, and not relevant to the instant discussion of understanding GAA, so my caveat may be out of the scope of your post.)
I had to take a break before I responded to this one, because reading this made me really angry and my first instinct would not have been appropriate.
If you think that this is a difference between you and the “G&A” folks, then you are not understanding us, because we don’t care about extra descriptive details either. “I pick the lock with thieves’ tools” vs, “I carefully insert the pick and lever into the lock, pushing down lightly with the pin to test the resistance of each of the tumblers, then gently apply just enough force to lower and lock them into position, one at a time, until they’re all in place and turn the lever” doesn’t make a difference to me. “I Attack the orc with my longsword” vs. “I lunge at the orc, feinting high, but then turning my blade low at the last moment, trying to get past his guard and jam the blade into the gap between his fauld and his grieves” doesn’t make a difference to me. “I cast fireball at that area” vs. “I give my wand a swish and a flick as I say ‘Incendio’ and point it at that spot, causing flames to erupt in a 20-foot radius around it” doesn’t make a difference to me. What makes a difference to me is “I try to read him to see if he’s lying” vs. “I watch his face for micro-expressions to see if I can figure out what he’s really thinking.” I care about clarity of intent and action, not detail.
From my standpoint, I don't see a qualitative difference between your fourth example and the first three:
- "I pick the lock with thieves’ tools"
- "I Attack the orc with my longsword"
- "I cast fireball at that area"
- "I try to read him to see if he’s lying"