While I think the "goal and approach" meme if overblown, I can easily put it in those terms.
It’s not a meme, it’s the actual way that the people who’s DMing style you claim to be trying to understand determine how best to resolve actions.
The declaration of insight tells me that the PC has the goal of determining whether someone is being deceptive with the approach of studying their words and non-verbal clues closely.
Except that, no, it doesn’t communicate that. The Insight skill can be applied to many actions which involve “gleaning clues from body language, speech habits, and changes in mannerisms” to “determine the true intentions of a creature.” So, if your goal is to determine whether or not the NPC intends to deceive you by carefully studying his body language, speech, and mannerisms, great. Tell me that, and I might tell you that you notice something in his body language, speech, or mannerisms indicate intent to deceive, or I might tell you that you don’t notice any such thing, or I might ask you to make a Wisdom check, depending on whether or not he does intend to deceive you, how he is or is not telegraphing that intent if he has it, and what might or might not happen if you fail to pick up on any telegraphs he might be giving. Asking for an Insight check is asking me to skip that process and just assume that you do have a chance of succeeding, chance of failing, and consequences for failing, and those aren’t assumptions I am comfortable making.
Does anyone ever really have the following conversation?
Player: I attack the orc with my longsword.
DM: Okay, give me an attack roll.
Yes. Happens at least once in pretty much every combat I run. There are certainly times when we shorthand it - if you’re within 5 feet of an orc, you have a longsword drawn, there are no other hostile creatures within 5 feet of you, and you don’t declare any movement before rolling a d20 and shouting out a number, yeah, I know what you meant. But if you’re standing there, 30 feet away from the melee, with 5 orcs all within range, and you just say “sacred flame,” I’m gonna ask who your targeting. If you’re surrounded by orcs, carrying multiple weapons, and you just say “15 on my Attack roll,” I’m going to ask you against who and with which weapon.
The activity being resolved is the insight check. If the NPC is using deception, how would you ever
know to ask for an insight check unless the players indicate that they are suspicious?
I don’t. If the NPC is lying to the PCs, I roll Deception for the NPC with the DC set by the highest passive Insight of the PCs being lied to. If he fails, I will narrate a tell, such as the NPC stammering, or sweating, or making shifty-eyes. The players are free to interpret that as they will. If any of the players want to follow up with an action, they can do so, announcing it in terms of their goal and approach, e.g. “I want to find out if he’s lying. I press him further, and watch closely for any signs he might be lying.” If they do, I will decide if this action succeeds, fails, or requires a check to resolve, and proceed accordingly.