In order to apply kotegaeshi in the right way, do you have to hold uke's wrist? Or do you have to hold uke's hand, placing your thumb on the back of his hand? I have seen it performed in these two different ways and I was wondering if there was a safety reason or a technical reason behind these two different applications.
Kotegaeshi (小手返し etimology) is a supinating wrist lock and is generally translated as "wrist throw". The throw works on the manipulation of the wrist, which turns the fore-arm, then the shoulder, then the whole body.
If tori's hand is supporting uke's wrist, then the twist will be much lessen. This means that there is less pressure/pain on the wrist itself. This can make the technique feel less horrible for beginner. However, it makes taking tobu ukemi (flip escape, 飛び受け身) much harder! Instead of flipping over uke's own arm, uke will tend to spin sideways. This bad habit will make it much harder to escape a more vigorous throw.
Thus, all kotegaeshi (this includes tenkai kotegaeshi/shiho nage as well) should be done on the hand, not the wrist.
Note that some junior clubs use to teach kotegaeshi on the wrist because of kids' weaker joints -- references here and here. This might be where you might have seen it done on the wrist instead of the hand.
There is a (tiny!) animated gif that shows that tori grips uke's hand to throw them. This allows uke to do a nice flip over their own hand/arm.
It really comes down to your style and sensei. In my form we focus on Kihon, which in itself has multiple translations. The focus is on basic of the basics so to answer your question is that you perform a kotegaeshi stretch during warm ups. The idea was to help show thumb placement and strengthen and stretch the wrist. We place the thumb at the base of the pinky.
Here is where it varies. The basic is to put your hand over your thumb. If you are throwing tobi-ukemi depending on size of the uke and a number of things, I have seen and done variations, but in the end the thumb is always at the base of the pinky.
Ask you sensei.