My understanding of kotegaeshi is nowhere near mastery. So, take what you will from my explanation. My disclaimer is that this is all my view at this point in my training. Things change, I'm not always right, and I continually learn new things.
The principle of kotegaeshi isn't to twist the wrist and make uke fall. Anyone with any understanding of relaxation and dispersing of energy will make this kind of kotegaeshi fail every time. I stand on one leg, lean over backwards and give my students my wrist. They attempt kotegaeshi through twisting of the wrist and fail most every time. The times that they don't fail are because I'm not great at relaxing and energy dispersion.
So, what is the principle of kotegaeshi? For most purposes, the "external" version is the common one and I'll describe my thoughts on that. Besides, I'm not that great at the "internal" version so I'd mess it all up anyway.
What goes on in kotegaeshi? Well, the first thing that happens is that we must control uke at the point of contact. In this case, our point of contact is the wrist. What do we want to do with uke? We want uke to eventually twist in an outward, downward, spiral manner. Now, there are many ways of accomplishing this and that is where we get into variations.
At point of contact, which is uke's wrist, we want to roll uke's center forward and twist/turn it. This causes uke to over-commit at the end of the attack. Uke will readjust to regain center/balance and in certain conditions, we can start to apply kotegaeshi. (In other conditions, uke's movement is in a completely different direction and kotegaeshi can not be easily applied. These instances call for other techniques.)
While uke's movement to regain balance is taking place (either hips shoot forward under shoulders or shoulders go backwards over hips), we roll uke's center backwards, twist/turn it, and spiral it downwards. Sort of like taking a screw and screwing it into a piece of wood, only done in 3 directions. In addition, this is done smoothly, such that uke's center rolls/twists forward and keeps rolling through to the outward and downward spiral. Principle wise, though, the main focus on kotegaeshi is that last part with the outward and downward spiral. You can be in any situation (ikkyo, gokyo, etc) and if you somehow find yourself at that part, then you've gotten to kotegaeshi.
An important point to remember is that we are not allowing uke's center to stabilize. For example, if uke's wrist is at mid-thigh level after the initial attack, we wouldn't bring uke's wrist up to mid-chest level to apply kotegaeshi and allow uke to regain balance. The end result is to have uke on the mat, so why would we allow uke to stand up and regain balance? Bringing that wrist upwards allows uke that chance. Now, note that I'm talking about tori/nage bringing uke's wrist upwards. That doesn't count for when uke is bringing his/her own wrist upwards. Those fall under variations and as long as uke's center is controlled and kuzushi is kept, tsukuri and kake should follow. Another view is how can you get uke to the ground if you are helping them stand up by raising their wrist?
At every point in time, we are concentrated on uke's center. The wrist is merely the physical link to the arm, which is the link to the shoulder, which is the link to the body, which leads to the center. Kotegaeshi is never about the wrist, but always about the center (which is the principle for every aikido technique, but that's another topic). In kotegaeshi, the center is turned in an outward, spiral, downward direction. The wrist just happens to mimic the center most times. But if you focus on the wrist, uke's center can do whatever it wants.
How do we accomplish the above? Physically, we use our hands to attach to uke's wrist. In what manner? That is where the variations occur. Depending on uke's movement, one variation might actually be more appropriate than the others. But, overall, it really shouldn't matter (except for each school's official curriculum for testing) where the hand is placed. In fact, if one has complete control of the center and great timing and excellent tsukuri, then kotegaeshi done with just the pointing finger of each hand can be accomplished.
Variations are great things. Where else can you work on a specific principle and do it from a multitude of physical differences? When beginning training, yes, we learn kotegaeshi from a specified technique. We're learning how to move, how to find the right timing and how to find the right tsukuri. At some point in training, though, we should be working on principles and not get caught up in what is the correct placement of my hand.
Again, all IMO, anyway. YMMV.