Ok. Consider combat as an analogy. In order to properly resolve an attack in combat, one needs to know which target the character is attacking and with what weapon, which can be seen as analogous to the goal (kill the goblin) and approach (by attacking him with my sword). Any additional details describing the specifics of the attack may be an enjoyable bit of color, but are unnecessary to resolve the action.
There are many situations where I don't need anything explicit. I mean, I do care but I know that Sly will always attack with their short sword and if the first attack doesn't hit they'll decide whether to attack with their dagger unless they've specified otherwise. We use minis so if Sly is indicating movement and moves into position that they can only attack 1 creature, that also does not need to be stated. Even if there are multiple targets, if the target they attacked last time is the creature is still around I'll assume that's the target.
Outside of combat it’s the same. When picking a lock, presumably your goal is to open the door/chest/whatever, and your approach is to pick the lock with your thieves’ tools. Any additional description about you carefully testing the lock pins with your probe or whatever are an enjoyable bit of color, but are unnecessary to resolve the action. When checking an area for traps, finding out if there are any traps there is a goal, comparable to killing the goblin getting a locked door or chest open. In order to resolve the action, I need to know what your approach is, Are you just looking with your eyes? Are you touching stuff with your hands? Are you using a tool of some kind? I need to know these things to determine what the potential outcomes will be, just like I would need to know what weapon you’re using when attacking or if you’re using thieves’ tools vs. smashing the lock or whatever. Checking an NPC’s emotional state for signs of deception is similar to checking a physical object or area for traps, just in a social context. Specific details about how you’re doing it are a nice bit of color, but are unnecessary to resolve the action.
From that example, I would go with option D. “I’m hearing that you want to find out if the chest is trapped. What does your character do to try and find that out?” If the player said they don’t know, I would offer some suggestions, like “you could try running your fingers or a tool of some sort over its surface to notice any irregularities, try and tilt it to check underneath, stand behind it and open it a crack to see if anything happens, or anything else you can think of, it’s up to you.”
So even with option D, you would still have them go through the motions of checking for a trap even though there is none?
As for the rest - maybe I'm just having flashbacks to the DM who would say something along the line of "Aha! You touched it! Make a constitution saving throw because you didn't look for contact poison first!" It became a game of player description versus DM gotcha. Now, I'm not saying that you would do that. However it's not my preference because it stresses player's skillful description over PC capability. I want PC proficiencies and how the player built their character to matter outside of combat. If someone wants to min-max their PC so that their fighter has maxed out strength and con while ignoring every out-of-combat skill, I think it should matter.
As far as stating specific goals and whatnot, for me it would just be superfluous most of the time. Obviously they're checking the chest for traps because they want to open the chest. Why else would they? I'm going to assume the rogue has thieves tools, just like I'm going to assume that Sly is going to use their short sword in melee combat and is not try to kick the goblin in the knee. I don't need to police my character's possessions and I assume that if they have the tool for the job they're going to use them. Why would they not?
There are exceptions to all things of course. If the monk is trying to pick the lock because the rogue is not available I'll verify that they have the tools and are proficient. There are times when I have to clarify exactly which creature the PC is targeting in combat or what they are trying to achieve. But 90% of the time or more, what they want to achieve and how they're doing it is going to be clear.
I think a big part of why I choose to focus on PC skill and not player skill is twofold. First, focusing on player skill rewards experienced players, particularly players who have been playing with a specific DM for a long time. Second, some people would find it very difficult to describe that kind of interaction either because they don't have the imagination, can't easily put it into words, have no clue what you want or just don't have the kind of personality that lends itself to that style.
Last, but not least, I fully acknowledge that I'm not the right DM for everyone. No DM can be. Most players seem to enjoy my DMing.