The medieval authorities disagree about whether this is factual or halakhic. I’m trying to understand the basis of their dispute: why should cooking be a “halakhic category” and not a matter of reality?!

You choose a minimal level of doneness that people would eat, and then check whether water in a secondary vessel can cook the product to the level of doneness we defined.

To go in the direction of saying that there is some abstract category of cooking that the Torah obligated us regarding, and we don’t know how to define it—that comes out complicated, and sometimes even absurd.

The Torah was given to human beings, and they can know whether there is cooking or not.

Because there are different levels of cooking, and there is no objective way to define the level of cooking required. So we formulate boundaries based on reasoning, and that is a halakhic line.
Try thinking about how to define concepts like baldness (how many hairs) or a heap (how many stones). There is no objective way to define them.
Indeed, the Torah was given to human beings, and precisely because of that, human beings define its boundaries.

The Torah was given to people, not to laboratories

How do you cook?
Every housewife knows that when you put a pot on the fire, a primary vessel is cooking.
A secondary vessel is already more like heating and less like cooking.
And that’s what matters.
What’s the temperature in the laboratory?
No housewife knows, doesn’t care, and it doesn’t matter to her.
What matters to her is that it should be cooked, and that’s in a primary vessel.

Interesting that some of the medieval authorities, the Hazon Ish, and the stringency of “the hand recoils from it” in Chayei Adam come to say that a secondary vessel cooks—we can see it—and the medieval authorities who disagree are basically disconnecting from reality and giving a conceptual argument. And I’m asking why. Go out and see, or as you said, Boris, ask the housewife..

Just for general information, I’ve seen quite a few cooking channels, and the logic about hot walls / cold walls is pretty accurate. Many times, a drop of 10 degrees in temperature damages the ability to cook. I saw a video demonstrating frying/cooking something (cuts of meat), and when they started adding it to the oil/water, the temperature dropped from 98 to 92, and that itself already required stopping adding more pieces to the pan until the temperature rose again, otherwise the pieces wouldn’t cook. You have to remember that the main cooking is inside the piece of meat, not just searing the outer layer of the cut. And when you transfer boiling water from one cup to another, the temperature will drop by much more than 6 degrees. The drop can reach up to 20 degrees. We don’t feel the difference because the difference between 98 and 78 is not significant in terms of the immediate pain we’d suffer from the burn. But in terms of preparing food, it’s very significant.

The easiest test is cooking an egg. If we take an egg and pour it into water in a secondary vessel, will it cook, at least to the state of a soft-boiled egg? Of course, that depends on the amount of water relative to the egg, so that the water’s heat is maintained; on the temperature of the egg; and on the secondary vessel—how much heat it draws from the water. Just for the record, egg yolk starts to coagulate at 65 degrees.