I agree with Greenstone.Walker's call here. An Intelligence (History) check or other "knowledge skill" is generally for recalling lore. Perception is for noticing a thing. Investigation is for deducing something given clues in hand.
I try to get my players into the habit of saying "I try to recall if I know..." or "I try to notice..." or "I try to deduce whether..." so that we have a consistent way of signalling a particular approach. Then I can decide if the character automatically succeeds, fails, or whether a check is in order. For example, I may decide that the aarakocra pirate character Beakbeard automatically succeeds in the deduction that the tattoo indicates the NPC to be a sailor. The criminal rogue Chuck Dagger may or may not make the deduction and makes a check. The hermit monk from the inland mountains, Ivana Stavemoff, may just fail to deduce anything.
Heh, well now we come to the whole "how exactly should I use the knowledge skills" thing. Different tables treat them in all sorts of different ways. In my case, I don't limit them to just remembering things. Let me try to explain my point of view better.
The reason I have a problem with Investigation is that "deduction" isn't really a skill all on its own. You can't train yourself in figuring out what any given clue means. People investigate by applying relevant knowledge to the object of interest. Any time you make a deduction, you're drawing upon things you already know, and finding connections to discover new information. As others have said, Wisdom makes you aware of potential clues, and Intelligence lets you make sense of the things you notice.
So the process of investigating something would depend on 1) how sharp your mind is and 2) how much you know about that something. To me, it seems the best way to handle that in-game would be to have Investigation just be a pure Intelligence check. The proficiency bonus implies specialization. You add it to the Int check if you know a lot about a topic relevant to the investigation.
Pure deductive reasoning and the like is the raw, cognitive power behind every Intelligence check. Intelligence
skills represent the things you've learned how to do using that cognitive power. It's just like how Charisma is the pure strength of your personality, and the various skills are different things you can do with it. There's no Charisma (Influence) skill because influencing others is what
every Cha skill does. That's the way I see Intelligence (Investigation).
The books never describe it as anything more specific than processing information about your surroundings, but processing information is what
every Int skill does. So when I try to work out what it's for my brain just... shuts down.
I liked the idea someone proposed about tying it to things you figure out about objects, and tying Perception to living creatures. That gives it direction. Ideally, I'd like to just replace it with a different skill entirely if I can't figure it out. It seems like thinking about it has only confused me more... >: