Start with what they secretly agree on, because it's almost everything.
Both camps want a child who feels safe. Both want a child who can eventually regulate. Both, practiced well, hold the limit. Neither camp's real ideal is a house with no rules, and neither is a house run on fear. Developmental psychology has a name for high warmth and high structure held at the same time: authoritative parenting (Diana Baumrind's work, later mapped by Maccoby and Martin onto two axes, warmth and demandingness). It's the quadrant the research keeps pointing at. And it's the quadrant both camps, at their best, are already aiming for.
So the fight is not warmth versus limits. Both want both. The disagreement is hiding in one word.
What is a boundary?
To one camp, a boundary is a rule you place on the child, with a consequence if they cross it: if you hit, you lose the tablet. To the other, a boundary is a limit you hold, about your own action, stated warmly and kept by you: I won't let you hit. I'm going to move your hand. Same word. Two different objects. One is a demand delivered to the child. The other is a commitment the parent keeps. Almost the entire argument runs on this single equivocation. Change which definition you're using and you change which camp you're in. That's the load-bearing belief. Pull it out and the fight falls over.
Underneath the word sits the one genuine question left: does a warmly-held limit, without an imposed consequence, actually produce a child who takes the limit inside, or do some kids need to feel a real cost before it sticks? That's not a slogan. It's testable. And self-determination theory (Deci and Ryan) gives the cleanest read available: structure a child can understand, with their perspective acknowledged, gets internalized better than the same structure enforced by pressure alone. The limit still has to be held. It just travels further when it isn't delivered as a punishment.