This is the clearest interpretation I've seen yet. So... for those of us who aren't using the alternative skill rules, it would come out to:
Perception: Attempt to notice a living thing that is hidden from you.
Perception, Passive: Sets the DC for a check made by a creature attempting to hide from you.
Investigation: Attempt to locate a hidden object.
Investigation, Passive: The highest finding DC you can pass automatically to notice a hidden object.
Yup. You could use my split of Perception (hidden living things) and Investigation (hidden inanimate things) even without doing the WIS / INT variant. So here's how it would look...
If a person, animal, creature monster etc. is hiding somewhere out and around you... you always have a default "spidey sense" passive perception to notice them of 10 + WIS modifier (plus your proficiency bonus if you have proficiency in the Perception skill). Anyone who is out there has made a DEX check (plus Stealth if applicable) to try and gain the Hidden condition-- your Passive Perception score is the DC they need to reach to avoid being noticed out of hand.
If the person's DEX (Stealth) check is higher than your Passive Perception... they are Hidden from you. On your turn, you can use your Action to make an Active Perception check-- Roll a d20 + DEX modifier (plus your proficiency bonus if you have proficiency in the Perception skill). Obviously, at the barest minimum you want to roll above a 10, since that's the only way to reach a number you haven't already gotten via your passive perception. If your WIS (Perception) check reaches the DEX (Stealth) check of the person Hiding, you have noticed where they are and can target them.
For traps and secret doors, using INT (Investigation) is the same thing. Walking around, a character has a passive sense to notice weird markings, objects that seem out of place, scuffs, scrapes, glints of light etc. This is your passive investigation and is 10 + INT mod (plus prof bonus if proficient in Investigation.) Any trap or secret door/panel that has a DC less than your passive investigation is noticed automatically. That tapestry that covers the secret door bulges where the doorknob is and isn't really hidden well at all. No need to roll-- you find it automatically. But for other things-- that pit trap which is built extremely well and not found via a simple tapping with a 10 foot pole-- you only can find it by using your Action to make an Active check, which is a d20 + INT mod (plus prof bonus if proficient in Investigation.) If you roll high enough to surpass the DC, then you find it. And in the case of traps or locked secret doors, once you have found it, it will then take a DEX (Thieve's Tools) check to disarm / unlock it.
That's all there is to it.
Now that being said... here's a few other reasons for what I personally do to add onto or explain this. The reason I use the Variant rules for skills is because I go back to the perhaps very first playtest packet, where they explained why they were using the six ability scores for checks. In that original document explanation-- the six ability score were and ARE the checks. In the way they were designing the game... they weren't having people make Perception checks to find something, people make WIS checks to find something. Likewise, to unlock a door you make a DEX check (not a Thieve's Tools check). To jump a gap is a STR check (not an Athletics check). To convince the shopkeeper to give you a discount is a CHA check (not a Persuasion check.) So in that manner of looking at the game... I see any passive "spidey-sense" noticing of weird sounds, out of place objects, strange reflections or whatever-- things that just tickle you at the base of the neck that tells you something is wrong... all of those are WIS checks. By the same token, any time you make an active search to find something, using clues and deductive reasoning-- that uses your INT.
Then on top of all of these six different checks... everyone has those few things they have a bit more knowledge about-- perhaps physically, intellectually, or spiritually. Those things (the things you have Proficiency in) can add a bonus to any of the six checks you are making. And they do not have to line up one-for-one. If you have proficiency in Religion... sure, to recall lore about a god would be an INT check (and you could add your prof bonus to it.) But if you were talking with a high-ranking church official and wanted to convince them to make a ruling by using some esoteric doctrine you know about, you could make a CHA check (plus your Proficiency bonus for having Religion.) Swimming is usually a STR (Athletics) check. But if you are swimming through a long underwater tunnel, I might ask you to make a CON check to hold your breath (and let you add your prof bonus for Athletics to exemplify your better ability to swim faster.) Or if you are trying to convince someone you are someone else, ordinarily that would be CHA (Deception). But if you had a illusion over you to make you seem like an ogre for example, you could attempt to fool someone by trying to lift a large object and make a STR (Deception) check.
These are all examples of why I prefer using the Variant rules. I always ask for Ability Checks, and the players can then try and get me to allow them to use a specific proficiency they have (if they can illustrate to me how and why it should apply.)