It's all about immersion
This part of your question jumped out at me:
I can see two ways of expressing the relationship between players and their PCs: either they act in place of the characters (“I'm Dunwick, I do this…”), or they control them like puppets (“Dunwick does this…”). I've never intended to impose a point of view or another on the players, but it looks like Dungeon World wants me to.
You're right, it absolutely does. Dungeon World has a very definite point of view. "Acting" your characters, immersing yourself in them, diving in as deeply as possible, is part of that point of view.
What does it mean to say that an RPG has a point of view?
All games have two components: a goal and a set of rules or constraints. Or to put it another way, what you want and how you're allowed to try and get it.
Some games have simple, definite goals like "score the most points" or "checkmate the opposing king". RPGs by their nature have looser, more open-ended goals that are partly determined by the players themselves: things like "get a ton of loot" or "unravel the conspiracy" or "have fun role-playing conversations between my character and other characters".
But that doesn't mean that all RPGs support all player goals equally. Even where the rules of an RPG don't make a certain player goal impossible, they can still make it more difficult, boring or annoying to achieve then other goals. (You could try to play a traditional heroic knight-like character in a game of Paranoia, but you'll be fighting the game the whole way.) Some RPGs try to support a wide variety of player goals. Others have narrower goals in mind, and their rules are designed to push players towards those goals. I call this alignment of goals and rules the game's point of view.
So when I say that Dungeon World has a very definite point of view, what I mean is that it has very definite goals in mind, and its rules are designed to strongly push the player towards those goals.
What is Dungeon World's point of view?
Fortunately we don't have to guess what goals the game designers had in mind. The SRD tells us up front:
Why Play Dungeon World?
First, to see the characters do amazing things. [...]
Second, to see them struggle together. [...]
Third, because the world still has so many places to explore.
The GM's Agenda section is also enlightening, and also names some goals that the game designers explicitly rejected:
Your agenda makes up the things you aim to do at all times while GMing a game of Dungeon World:
- Portray a fantastic world
- Fill the characters’ lives with adventure
- Play to find out what happens
Everything you say and do at the table (and away from the table, too) exists to accomplish these three goals and no others. Things that aren’t on this list aren’t your goals. You’re not trying to beat the players or test their ability to solve complex traps. You’re not here to give the players a chance to explore your finely crafted setting. You’re not trying to kill the players (though monsters might be). You’re most certainly not here to tell everyone a planned-out story.
If I had to sum all this up in one sentence, I would say that the goal of Dungeon World is to immerse the players in a collaboratively-generated adventure story of vivid characters interacting in a fantastic world.
The rules of Dungeon World are all designed to push the players towards this goal in one way or another. For instance, the "collaboratively-generated" part is encouraged by severely restricting the GM's ability to pre-plan campaigns, giving the GM moves that can generate new plot points on the fly, and giving the players the power to push the GM to flesh out parts of the game world that the players find interesting (through moves like Spout Lore and Discern Realities). All this discourages railroading and encourages shared improvisation.
What does all this have to do with addressing characters instead of players?
The real question is, how does this rule help to push players towards the game's goal?
Let's start by reading this part again:
If you talk to the players you may leave out details that are important to what moves the characters make. Since moves are always based on the actions of the character you need to think about what’s happening in terms of those characters—not the players portraying them.
So it's not just about doing a word search on your sentences and replacing "Tony" with "Dunwick". It's about forcing yourself into the mindset of the characters. What do they know? What can they see and hear around them? How does that drive their actions? If you are actively thinking this way, you might realize that you forgot to mention the lead goblin is wearing extra armor, or that the friendly shopkeeper is a Tremarian (Dunwick is racist against Tremarians)...or on the other hand, you might realize that the archer you were about to mention is sitting motionless in the shadows 100 feet away, where Dunwick would have no chance of spotting her. You the GM become more invested in the story just by thinking this way, and with fewer hiccups and less metagame-y infodumping, the story becomes more believable for everyone else.
But even if it was just a simple word-swap, it would still help the players get immersed in the story. Imagine watching a play where the actors keep saying each others' names instead of the characters' names. Changes the mood, doesn’t it? It emphasizes the artificiality and "play-ishness" of the play, which puts a distance between the audience and the story. Dungeon World doesn't want to create that kind of gap. You and the other players are actors in a sense; you are all performing your part of the story for each other. Using the players' names is a distraction. Using the characters' names, on the other hand, generates a continuous string of little near-subconscious reminders to focus on the story and characters, which is what the game wants.
What happens if I break this rule?
Not much, honestly...assuming you are only breaking the names part specifically, and are otherwise doing what you can to preserve the story-focused frame of mind that the rule encourages. (Thanks to Dave for pointing out this nuance.) Alex P is right, this rule by itself isn't a huge one, and leaving it out won't kill the game...but by the same token, leaving it in won't kill you. Why not give it a second chance? Now that you understand some of the reasoning behind it, it might grow on you.
What if I don't even want to immerse myself in the story?
If you and the game disagree on goals, you're going to have a bad time. I don't think this is your case, but if it is...Dungeon World may not be the right game for you.