In aikido, I was taught to hold the very wrist when performning kotegaeshi.
In another art, I have been taught to just hold the hand - not the wrist, since that would make the technique less efficient. So they say. If pain in the joint is what you're after, their point is obvious. Holding the wrist stabilises it, supports it. But other than painwise, I wonder. What aspects - if any - of kotegaeshi is better performed holding the wrist, compared with just the hand?
Would the standard aikido way of performing kotegaeshi as a throw, with uke taking high falls, be possible holding the hand not the wrist? Is holding the wrist instead of the hand a safety measure?
How do the DR people perform their kotegaeshi?
Kote means "wrist", for sure. But even if you hold the hand, it is the wrist that is twisted - actually, even more so.
You question indicates that when somebody is pointing you a moon you are looking at his finger.
HOW/WHAT you grab in kotegaeshi doesn't matter at all. What is important, is to create a connection to his center (i.e. by locking all joins from wrist to hips) to be able to unbalance him. Sometimes it is enough to use only his small finger. Sometimes you can use his elbow or shoulder…
If you are interested only by his hand/wrist, you will lock only wrist. If he is not unbalanced, you can break his wrist but you can't throw him.