An investigation check is most definitely not just a more specific perception check at my table. I mean, for one thing I call for the ability and let the player determine if one of their proficiencies applies, so it’s really just a wisdom check, to which you can decide if you think your perception or investigation proficiency (or some other proficiency or no proficiency at all) applies. But, by default the investigation skill is a specific application of the intelligence ability. According to the PHB, “an Intelligence check comes into play when you need to draw on logic, education, memory, or deductive reasoning” and proficiency in the investigation skill applies “When you look around for clues and make deductions based on those clues.” In contrast, Perception is a specific application of wisdom. The PHB says “A Wisdom check might reflect an effort to read body language, understand someone's feelings, notice things about the environment, or care for an injured person” and perception proficiency applies when you try to “spot, hear, or otherwise detect the presence of something.” In other words, Perception is for finding the clues, investigation is for deducing their meaning.
I don't think I'm explaining myself well.
If I say "My character goes over to the bookshelf and runs my fingers over the books" I'm investigating. That is what an investigation looks like. I'm not perceiving. Perhaps I pick up a mug and examine it in close detail. That's investigating. I'm getting in close. This is why the Eyes of Minute Seeing give advantage to Investigation checks, while improving vision up to a foot away.
So, if anything where I get close, touch things, examing in minute detail is investigation, then perception is just... looking. The rules text you just posted says it, "notice things about the environment" that's perception. Again, "spot, hear, or otherwise detect the presence of something." With that in mind... what action do you take to trigger a perception check, except the action of "looking" or "noticing"
I mean, you would never accept someone stating "I notice the goblin hidden in the corner" as their action. In fact, since the goblin was hidden and they didn't know about it, they could not make that statement. But noticing is the action of perception. This is the fundamental problem I have with the repeated assertion that a player saying "I look around" isn't good enough for a perception check, because... what else are they supposed to do? There is no other action for visual perception. The only other conclusion is you want them to narrow the scope, to only attempt to perceive part of something, which leads into my other concern, as I stated.
I don’t see anything unreasonable in that exchange.
Well… Except I guess for the part where the assassin successfully stabs the PC in the back without initiative being rolled or an attack roll being made. But I don’t think that was what you were meaning to use the example to illustrate.
I do see something unreasonable in that exchange. Because the player was clearly trying to use their action to detect a threat, but because they didn't know where the threat was, they automatically failed. However, you can't know where a hidden threat is until you spot it. It basically turns perception into extreme three-card monte, where you walk up to someone with three face down cards who has already shuffled them. It is just blind guessing.
Which leads to one of two responses from the players
1) They will list every single thing possible, turn over every card, in an attempt to win, This is undesirable, because I do not want to sit through your checklist every time you enter a room.
2) They go vague, and ask to roll perception. Understanding that this fundamentally covers everything they could potentially try and guess, with a single action.
Yes. From them holding up the die and saying “religion?” I can reasonably surmise that they want to know something about the idol, and that they think their proficiency in religion would be applicable if a roll is required. But they have not conveyed any information about what they want to know, nor have they provided me with enough information to determine whether the attempt to learn that information could succeed or not.
They want to know the things they don't know about the idol.
This is a fundamentally strange question to me. Let us say this is an idol to Shar, but it is made of purple amethyst, which is an important detail because it connects them to a specific cult of Shar. The player asks who the idol is of, and you let them roll and find out about Shar.... but since they didn't ask about if the material was important you aren't going to tell them? They have no idea to even ask that question, so why would you expect them to ask it?
Again, this seems to lead into blind guessing games. Can you ask the correct question to unlock the hidden information you don't know to ask for? And I don't see the appeal of this.
Asking what about the idol though? Are they asking if they’ve run across it in their studies? If so, they could simply say that, instead of relying on me to correctly guess that instead of making a different assumption. Maybe I think they’re asking if it depicts a god they’re familiar with. Maybe I think they’re asking if it displays any signs of having been desecrated. Point is, I can’t read their mind, nor am I interested in trying to guess what they meant.
All of those. They want to know all of those things. All of them might be important.
Why do they need to specifically ask if the idol has been desecrated for you to provide them with that information? The idol is entirely in their mind's eye, they can't see it and compare it to a non-desecrated version they have seen before. Which in IRL they would obviously be able to trivially do. I don't need to specifically ask out loud if a cross has been desecrated, I'm pretty sure I'd be able to tell by looking at it, because I know what a cross looks like to begin with.
I’m not asking where they acquired their proficiency, I’m asking where they imagine their character might have learned the information they want to know about the idol. Doing so gives me something to assess to determine if the action can succeed or fail and what check to call for if both are possible, and it reveals an interesting detail about the character’s backstory.
You aren't asking where they acquired their knowledge of religion (where they got their proficiency), just asking them to imagine where they might have learned information about something they don't know to help reveal details of their backstory... how are those not the same thing? They are expecting they learned the information in the same place they got their other knowledge of religion, which is what gave them proficiency and training in the religion skill.
When you walk into a room, I describe anything you can perceive that isn’t hidden, and use your passive perception to determine if you notice anything that is hidden. That’s how the rules indicate that such general “looking around” should be resolved. If for some reason you suspect there is still something hidden that you missed with your passive perception, you have to do something if you want to find it, and you need to tell me what it is you do. Is what you do altering the environment? Will an Intelligence (Investigation) check be needed to resolve it? I can’t know until you tell me what it is you do.
But I want to "detect hidden elements of the environment" and there is no action for that except looking around and noticing hidden elements of the environment. Are you saying that you have never had someone roll perception? I can't think of an action to look around for hidden or concealed things, except to say I look around for hidden or concealed things.
Great, and for that you get to apply your passive ability (skill).
So what does an active insight look like? How can I actively use it instead of passively?
If the players think that me describing the NPC as unusually sweaty is enough to confirm that the NPC is lying, they are free to proceed according to that assumption. I generally advise that it’s smart play to take steps to confirm your assumptions, but that’s the players’ decision to make.
They didn’t miss any clues. I described what they perceived, as it is the DM’s role to do. They are free to decide what to do with that information. Again, I advise that it’s smart play to try and confirm their assumptions through action, but it’s up to them.
How can they confirm it? You refuse to let them use the skill to confirm it unless they "do something" but there is nothing to do, because the action that constitutes "an effort to read body language or understand someone's feelings" is being relegated to a passive skill that they cannot utilize.
What actions do you expect from them?