Perception versus Investigation
All of the examples with Perception involve hearing things and all of the examples with Investigation involve looking and touching things. This makes me think that calling for a check would really depend on how the player narrates their character's actions, but most of the time it's going to be Investigation.
The great utility of the passive check is to determine whether a character detects something secret without the need to roll dice and reveal to the player that something is there. I really like the concept of passive skill checks because it means I can flag areas of dungeons ahead of time for things the characters will detect just from walking down the hall or looking around the room for the first time. Also, Stealth checks are countered by Passive Perception, which can also apply to NPCs for when PCs want to sneak past guards or into jewelry stores. This gets into a sticky gray area when it comes time to have social secrecy...
Lying Liars and the Lies they Lie
Deception is a skill under Charisma and Insight is a skill under Wisdom, and according to what's been established on the previous page with Stealth and Passive Perception it would seem that Insight should be Passive and the one attempting to lie rolls their Deception. I've been having PCs roll Insight in some situations and now I'm beginning to think that if I want to maintain consistency that I should revert to the Stealth-Perception dynamic.
Since I started running 5th edition D&D I've had two different players in two different games convinced that an NPC was lying simply due to the circumstances in which the NPC was speaking. In both situations the NPC was not lying, and in both situations I approached the skills in a narrative way. In one situation, I had the player roll their Insight and they rolled above 15 so I decided the NPC was lying in accordance with what the player expected. In the other situation I asked the player what their Passive Insight was and it wasn't above 15 so I decided the NPC wasn't lying. Both situations were wrong.
In both situations I should have ruled that rolling above 15 means the character realizes the NPC was not lying, since the player was so dead certain that they were. Passive Insight should pick up obvious lies. Scrutinizing an NPCs words and body language should have an Insight check to determine the truthfulness of their words.
Get the hell out of my way!
Intimidation is a pretty straight forward social skill, but what does it mean in terms of combat? You can't bluff an Intimidation, because bluffing would fall under Deception. Intimidation requires the will to enact violence. If you fail your Intimidation roll, then you have to make good on your promise of violence. Consider the options:
You threaten someone and they resist, then you inflict violence. Your threat was meaningful.
You threaten someone and they resist, then you do nothing. Your threat was meaningless, it was actually a bluff.
(Sidenote: this is also my primary argument for why I use Manipulate for bluffing in Apocalypse World)
If your threat is meaningful then your target is really just trying to determine one of two things, whether you intend to act upon your threat or whether they can take you in a fight. In either case, the Intimidation roll would tell them either "Yes, he intends to act upon his threat." or "No, I don't think I can take him." and would best be resolved with a contested Perception.
Additionally, I think Intimidation should fall under Strength as a skill, since having an imposing frame (or a slight one) can greatly affect whether or not somebody thinks they can "take" you.
Buddy, can you spare a dime?
Persuasion falls into a strange category where it is a skill used to coerce behavior out of an NPC, but it uses neither subterfuge or brute force. Applying charm and good manners to convince somebody to take a course of action.
Stealth and Intimidation are countered by Perception
Deception and Persuasion are countered by Insight
That's my take on it.