I feel like Praion already answered your question.

So I'll expand: Draw Someone Out is the manipulation move! After the lord insults you (or maybe before you demand something), you Draw Them Out and ask "how could my character get the lord to swear fealty". Then the MC gives you the leverage you need to get in order to make him comply. No need to roll a manipulation move on him once you have whatever it is he wants.

Or at least that's how I read it.

Yeah, using Draw Out as a manipulation move is great. I like that move quite a bit, as mentioned in the OP, so the more I get to use it the happier I'll be. But I don't think it covers "well now I pull out a sword, last chance, buddy." There's something else going on there.

Hmmmm, maybe my question isn't really about Claiming, it's about escalation. Let me try to phrase it a bit differently:

If a character says "I pull out my sword and demand fealty," right away just like that, you'd probably go "Oh, you're Claiming Your Right! Roll for it!" Makes sense, yeah?

However, if she says "I remind him who my parents are and demand fealty," then you're also probably gonna say the same thing: "oh, you're Claiming Your Right!" This also makes sense, right?

And then let's say the lord goes "No way, you cowardly swine" like in our running example. At that point, the circumstances have changed. The lord has refused and is insulting the PC; time for the character to react again. If she wants to attack and make an example of the lord, it's easy to see how the rules accommodate that. Buuuuuut, let's say the character doesn't want to kill this lord, maybe doesn't even want to hurt him, she just wants to show him she means business. So now she's all "I pull out my sword and demand fealty for really real this time."

This is, fictionally speaking, a totally legit next step. The player is escalating the situation: now it's all about threats and her willingness to back it up. The lord could totally change his mind- steel is drawn! Things got heavy! Maybe he felt confident before but is now shaken. Maybe not, maybe he's still gonna say "you and your fealty can rot in hell."

But I'm not gonna ask my player to roll another Claim right after the first one; that's weird, right? So right now it feels awkward to me, this idea that a player can make a legitimate escalation and I'm kinda wondering how to accommodate that with the new Moves. That's what the real crux of this question is: can you escalate your demand after a refusal? If so, is it just the same Move again?

Does this make sense?

(edit: I'm not trying to move the goalposts on this, I'm really curious how this would resolve with the new Moves, is all. Thanks for the discussion so far! I keep thinking that maybe just using social leverage isn't Claiming Your Right at all. Maybe the move doesn't trigger until the character has at least implied retribution or consequences, and that's what really makes it a Claim. I'm not sure, though.)

That's legit. I guess it seemed a little like they were still able to ignore your claim. Maybe it might help if the trigger was a little more "make a claim you're prepared to back up"? I was mulling it over and thinking to myself, what if it's not triggered until the player implies repercussions? This might be in the wording of the move, or just made clear in the book and applied at the table. So, like:

Player: "I ask him to swear fealty."
MC: "he's all, 'sorry kid.'
Player: "I pull out my sword and demand it."
MC: "ahhh, you're Claiming Your Rights? Roll away..."

Maybe I was jumping the gun on the trigger? Maybe it doesn't happen unless the character is serious serious.

I really like the new Basic Moves, especially Draw Someone Out. But I'm missing that step in between violence and demand; where a player shows that they're serious but gives them a chance to acquiesce. I miss Aggro or Manipulate. Check this out:

Player: "I yell at him: 'you swore fealty to my father! You will not deny me now! Join my cause and maintain your honor."
MC: Are you Claiming Your Right? Sounds like it. Roll Hard."
Player: "Sure, it's an... 11! Bite it! I stare him down, fiery."
MC: "The wayward lord stares right back: 'Times have changed. Lords have changed. I owe nothing to you.' He's calling you out, but you get a +1forward.
Player: "Damnit. I draw steel and hold out my blade, like 'Bend your knee or I'll bend it for you.'
MC: "Are you willing to kill over this?"
Player: "Uhhh, I don't know. Maybe, but I don't want to. I want him to know I'm serious, but I want him alive to lead his men."

...so is there a Move that comes next? I would've previously used Aggro or Manipulate, depending on how far the player is willing to take it. They deserve a roll because they've got that +1forward. What happens next?

When the dragons of the earth give your player 2 armor, that's a great time to ask them what that means. Maybe they're like, "oh, the earth moves to protect me, like walls of stone" or maybe they're like "oh, I'm just lucky, I guess." Then you know what they think is cool and what they're after.

The book doesn't make it clear, but Vincent has said that when he runs battles (or other tense situations), he goes around the table asking for actions and then has everyone roll and resolve simultaneously. My group doesn't often get in fights together, and when they do I often ignore this advice out of habit. When it comes to announcing simultaneously...

Pros: everything is maybe more visceral, reactions are based on instinct and motive, it preserves the unpredictability of combat

Cons: players can't react to a success or failure until next "round," it could make a player feel like they're "giving up an action" if they Aid, it could lead to weird timing questions (though both methods run into this)

It's all preference, I think.

Yeah, make the firefight about the game fiction, with movement and goals and sudden changes. Remember that there's no "to hit" roll. A player can accomplish much more than "I shoot" on their turn. If you read the rulebook, one of the play examples has a player Seizing a bunker with a single roll. Not trading shot by shot with each enemy, but making one Seize by Force that encapsulates several minutes of battle by asking "what do you want? Oh, you want the bunker? Well, roll +Hard..."

As players, they can demand bigger rewards by setting the stakes higher than a single gunshot. Try pushing for a slightly bigger scale, like "okay, you're shooting and ducking, what's the ultimate goal here? You want him to back off the car, it seems like?" Or "...so you're trying to get him out of cover?"

If you just go through the battle trading blows, the system isn't the most exciting. Make Harm less mechanical and more scary. Make them realize that they are fighting for a reason, not just because. Have the battlefield shift and move and make their rolls consequential in ways that Harm is not, like cutting off escape routes or destroying nice things.

That's what I try to do when my players get into a fight. To me, it's not a great tactical blow-by-blow system. It's great for cinematic fights, so that's what I go with: lots of jump-cuts and zooming in and out.