Re: Kotegaeshi: hand or wrist?
I've seen it done many different ways, but that doesn't mean that each is equal or that it doesn't matter how you go about it. I would agree with Phi up above that the issue is about destabilizing the other person. How that is done via that particular connection can vary. Some emphasize strong bending of the wrist to stretch out the connection (take out the slack). Often they don't have to use the fingers to "finish things off" as the slack is all out already when done "correctly". Others affect the balance of the attacker by manipulation of the forearm more or less directly. And then there are variations in how much twisting is done, when it is done, how it is done, etc.
My point is that you can really screw up the technique by mixing aspects of various ways of doing it. Or you can do a very nice kotegaeshi by making sure you're doing all the "correct" aspects together. Meaning if you're on the hands and you're not doing a lot of twisting you need to get the arm more extended and bend the wrist more completely. And so on.
There are a lot of ways of doing things "right". And some of them are totally wrong *given* the underlying "operating system" of another style. And you can really mess things up by mixing and matching which is why many who start "innovating" well before they've even come close to understanding one way are often so terrible...