I check the chest for traps....etc.

I've been playing D&D since the Red Box, and I occasionally find myself with a major case of Edition Drift where rules are concerned.

So, please help me out here.

How does checking for traps (in a corridor, room, chest, door, etc.) work?

If the party is simply working their way down a corridor, and they're all being wary, we can assume a Passive Perception roll is enough to notice things that might be out of sorts; tripwires, small holes in the wall/door, oddly raised stones on the ceiling/floor, etc? So, in this case, I simply compare the PCs' Passive Perception scores to whatever DC I've set for trap detection?
And this would also apply to just entering a room, or glancing at a chest, or any other simple observational interaction?

Anything more involved (hands on, a little testing, tracing of wires, and overall more focused) would be either the Study  or Search actions?

I see that Search can involve an active (normal) Perception roll to specifically find a "concealed creature of object."
Whereas Study can involve an active (normal) Investigation roll to figure out how a trap works. Which now begs the question of what a Passive Investigation score is for...

So is it sort of DM caveat according to the actual scenario as to what gets rolled if and when a Passive Perception score detects something "off" in the first place?

Really, the more I try to read "helpful" guides online, the more I find myself confused...

Typically, if the party members aren't actively going "I'm looking for things" you can use Passive Perception if you don't want to tip off the players (make sure you know them already) or ask for a Perception check (great for inducing paranoia). I think Passive Investigation is probably mostly for "you know enough about traps to figure out this one without rolling", but since it only shows up on D&D Beyond and not in the books, I'm not 100% sure about that.

If the player says that they want to examine something more closely, or look around specifically and take the time to do so, let them roll, but until then, I would use passive perception.

I use passive perception to see if players would get the feeling that something is off and would need to roll, or if they just don't notice, and sometimes if it's high enough they do just notice things, but I like using phrasing here- "The room seems empty" is more likely to get a player reaction than "It's an empty room". You can use this to encourage players to look closer, or try  to hurry them past things, depending on the situation.

Definitely going to echo the above advice about using Passive Perception. I've had DMs say out of nowhere "roll a Perception check" and I don't like it. That's literally what Passive Perception is for; seeing something without intentionally looking for it.

And I think you've got it right. Just ignore Passive Investigation. It never appears in the Player's Handbook and doesn't make sense, considering that Investigation is meant to be hands-on, active searching.