"Here’s a fun thing to do: “Keeler, this person named III corners you that night. She’s fucking pissed off, she comes straight at you, fists first. What did you do to her?”
Maybe Keeler’s player will answer with something. “Dude, sure, I’ve been sleeping with her guy.”

Here, Keelers player decides that her character has been sleeping with III's dude, but in a traditional rpg, a player wouldn't make up such a detail of the background, at least not in my experience.

My experience differs- I'd frequently see details in a character's background like this. What I don't usually see is it being made up during the game, it's usually part of the backstory written up before the game begins, and I think that's interesting.

In response to your question- the MC can ask the player's questions. They should probably be questions that can be answered in character, from the characters perspective. It's recommended that you ask questions like crazy during the first session, but besides that I'd say it's a matter of taste. If you've already created part of the setting you won't need to or be inclined to ask questions about it, but asking them to fill in details from their character's perspective is always an option that's available. I bet the balance that gets struck between the MC making the setting without asking a lot of questions, and answering a lot of questions with questions, is going to differ a lot from game to game. Ultimately I'd say the MC is responsible for describing the setting to the players, but I think asking the players questions is a useful tool for doing that.

What I'm finding a little confusing here is that from reading the book, and from Chris and Allison's answers, I get the impression that all I have to decide to know whether Manipulation is being used is whether the PC has and is using leverage to get something they wouldn't otherwise. That's a judgement call I feel comfortable making.
That would sometimes include negotiation. It's necessarily a conflict of interest over the original issue, but the manipulated party could walk away feeling they were better off if the leverage was a reward rather than a threat.
Noclue seems to be saying that there is more to it than that. But the only difference I can see between my version of the car negotiation and his(hers?) is that his feels... slimier. Now, I can see why you wouldn't want a move called Manipulation to be used in a way that isn't manipulative. But I'd bet if we asked six different people on this forum what the difference between just talking, persuasion, negotiation, and manipulation is, we'd get twelve different answers. Vincent and I may not agree, and I may not agree with the other players. "Does this feel manipulative enough?" isn't a judgement call I want to have to make every time someone wants to use the move.
So does manipulation require something more than leverage? Is it something with a clear dividing line, like leverage, or do I have to gauge whether something feels/seems like manipulation to me?

I would say that if you propose something we both agree is in both our interests, that's not manipulation. If you propose something that you feel is obviously in my best interest, but I resist doing it. If you coerce me into doing it, whether through preying on my greed, libido or fear, that's manipulative. Even if you tell yourself we're both clearly better off. Even if we're objectively better off by any metrics that make sense.

How can you coerce someone with greed or libido? Isn't coercion by definition the use of force or threats? Imagine someone tells you they were coerced into giving up a secret. When you ask how they were coerced, they say they were bribed, they were coerced through their greed. Wouldn't you say them accepting a bribe was totally voluntary, and calling it coercion would be inappropriate?

I think you are referring to the proposal at a different stage of the negotiation here than I was a moment ago. Let's consider the example with fixing the car from earlier.
Offer A: Fix my car for nothing.
As you say, if this offer is acceptable to both parties, there is no need for manipulation. Let's assume it isn't acceptable. I can use the fact that you want to drive my car as leverage, and use that to manipulate you into fixing my car. We can sum this up as a second offer.
Offer B: Fix my car, and I'll let you drive it.
Now, that offer might be so clearly acceptable to both parties that a roll seems unnecessary, if it's just completely obvious that you'd accept that. That doesn't mean manipulation hasn't occurred, it just means we didn't need a roll to resolve it. Suppose it's unclear whether it would work or not, and it would depend upon the finesse with which I made the offer, or brought to the surface your desire for the car. Then it might require a roll. Either way, if I successfully manipulate you into accepting the second offer, that must mean you believe you were better off accepting the offer than not. In other words, I'm not arguing that we were both better off with the first offer, clearly you didn't think so or manipulation would not have been necessary. But we could very well both agree that we are better off with the second offer. On the other hand, I could have taken a totally different approach.
Offer C: Fix my car, and I won't burn your house down.
You'd only take that offer if you thought it was better than not accepting it, but you certainly wouldn't say we both came away ahead on that deal.
I'd describe threats as social violence, but I wouldn't describe rewards that way. Manipulation includes both cases. 

It might be useful to differentiate between the Manipulation move here, and the broader colloquial sense of the word manipulation. It seems like you reject that anything which doesn't isn't manipulation in the colloquial sense could fall under the Manipulation move. Now, we might have a hard time agreeing to the meaning of the word manipulation, but the Manipulation move seems to be defined reasonably clearly in the game text as the use of leverage to get someone to do what you want. I think there are cases where we can use leverage to get someone to do what we want, and thus the Manipulation move applies, but it could be debatable whether it's really manipulation in the usual sense of the word. Just like the Seizing by Force move doesn't always involve seizing anything, and acting under fire is very broadly interpreted and covers cases that you wouldn't call acting under fire outside the game.

The more I think about it, the less well the idea that manipulation is always "social violence" sits with me. Violence entails harm, and the violation of rights. Using leverage can be violent when the leverage is a threat. But offering someone something they want is also leverage. Even when there is a real conflict of interest, such as when you bribe or seduce someone into doing something they wouldn't or shouldn't do otherwise, I wouldn't say you've harmed them. It seems entirely possible that the manipulated party is even better off than they otherwise would have been.

And I'd certainly say negotiation can be a form of manipulation. In most cases it doesn't involve a roll, but that's not because it isn't manipulation, it's because it's completely obvious that the use of leverage is successful. It's like how sometimes you roll for going aggro and sometimes you just inflict harm, or how you need to read a person sometimes but other times it's just obvious what they are thinking. Sometimes you say of course your barter is sufficient leverage to buy this pistol, that's clearly an acceptable deal to both sides so no need to roll. That doesn't mean that situation is fundamentally different from bribing someone to keep silent, it's just that in the latter situation it's much less clear whether the leverage is sufficient, and it needs to be applied with much more finesse.

I mean, when it's PC vs. PC and you're trying to figure out when to make it a Move or not.

Chris

Yeah, I was under the impression that you didn't need leverage when it's PC vs PC, so you don't need that to figure out whether to make a move or not. You use leverage to decide whether to make it a move or not when it's an NPC, I'm not sure what not requiring leverage for PCs would mean other than that you don't need leverage to make the roll.

In the example of two PCs on pg. 198 of the rules, Keeler just wants Bran to like her, and hits a 7 so she can offer him an experience. No mention of any kind of leverage. Maybe it just got left out of the example, but there is a thread somewhere where Vincent clarifies, maybe I can dig that up.

Edit: Found it!

For PCs, you don't need leverage. You get the carrot & stick instead.

This gave me the impression that manipulation with PCs is a whole different beast.

To actually address the original topic-
I've noticed in many games that intimidation is a terrible tactic. You announce your presence and intention of aggression to your target. Maybe you are still so overpowering that they choose to cave, but often choosing to intimidate makes you less intimidating, because you've given up the initiative. This move makes intimidation a much stronger option because if they choose not to cave, you still get off the first shot.
So one difference might be that instead of going aggro, you would manipulate with the threat of violence as leverage. When it didn't work, you'd be more likely to lose the fight because the other side would act first. Whereas with going aggro, the side that goes aggro is more likely to win if a fight results.

You can get 'do overs' or not pull the trigger.

Go look at the write up in the rule book, where Vincent explains Go Aggro.

Not sure what you mean here. The book explicitly says bluffing counts as manipulation.

Edit: Also "At this point, the player can't decide not to inflict harm, it's gone too far for that." So no "do overs" or deciding not to pull the trigger.

Wow, that's not how I took reading a person either. I thought that could just represent paying attention to other people and being empathic. Why does trying to observe what someone is feeling, or figure out what they want you to do, necessarily involve distrust or penetration of defenses?
There are a lot of reasons for not just asking someone. Being able to read someone's emotions and desires without asking them is often a sign of emotional closeness, not emnity.