Proper knife position--why?

Two main reasons:  1 ) You want the knife to do the work and not the downward pressure from your hand.  You are able to experience the difference very easily when cutting a tomato.  2 ) Over long periods of time ie 70hrs/week for 20 years  your hand will be so arthritic if you do not use proper technique.  After that amount of time they probably still will be but the severity will be less.  It is hard to train your hand from what you are used to, but it is well worth it.  Also shows how poorly sharpened your knives are

Another reason is, by pinching the blade with your thumb (on the left side) and your index (on the right side), you're holding it steady vertically. You're controlling the direction of the blade, and you know where it's going. If you just push on it from the top with the thumb, you're just exercising pressure on it, but not controlling its direction. It could start twisting to the left or to the right, making your cuts not straight, and risking cutting your fingers rather than your food. 

There are a lot of reasons to hold a chef's knife with a "pinch" instead of any sort of baseball grip; and there's no really good way to rank them in order of importance. 

A pinch keeps your knuckles to the side, and not underneath the handle where they can smash on the board.  A pinch gives you better control over the point of the knife.  A pinch acts as a fulcrum.  A pinch requires less effort.  A pinch locates the very back part of the edge.  A pinch 
"chokes" the length without hiding the heel, Etc., etc.

On the other hand a baseball or modified baseball grip provides more power and allows you to get more out of a dull knife.  If your knife is dull (and it's almost certainly very dull) a strong grip will feel more natural.

There's a little bit to getting a pinch grip right.  If you're willing to learn we can talk.

BDL

I'll just give a very intuitive  home cook's point of view, and my knives are inexpensive though i think fairly sharp. I have no awe of knives and don't enshrine them or endow them with special powers and don;t spend a small fortune on them.  I've never actually "sharpened" a knife, it seems, though i do use a steel pretty insistently and they are pretty sharp.  But maybe your chopping knife is not sharp enough or the blade is too thick.  Maybe that forces you to use another technique.   

It seems from your description that you are actually pushing the blade down and then lifting it up.  That means that the movement is using a lot more energy than necessary.  With all skills, the sign that you're doing it right is that it's effortless. 

The thumb and forefinger hold on the two sides of the large end of the blade is like a hinge.  The thumb and forefinger of the other hand on the point end of the knife is another hinge.  The up and down movement of the wide end (the right, if you're righthanded) is not a movement of the arm but a movement of the wrist.  If you grip the handle and put the thumb on the top of the blade, you're forced to use your arm. 

Arm movement is a much more tiring movement, using far more energy, is less accurate and is much slower  I can hit the board with the blade using the two pivot "hinges" many more times than you can because the wrist moves faster than the arm.  Imagine using a hammer with a 3 foot handle to hammer a small nail into wood.  It would be less accurate and would involve slower movements. 

Once i was fanning myself in the 100 degree heat and a colleague said he read some research that said that fanning yourself actually produces more heat (because of the energy used to fan) than is compensated by the cooling function of the hand. 

I know how that research must have been done.  They got some men to fan themselves and of course they used their whole arm to do it.  I guarantee that if you use your wrist, you produce much more coolness than heat!   

I believe the pinch method is best and I've adapted to it myself. That being said, I often go to friends houses and whether the pinch feels natural or not varies quite a bit depending on the design of the knife. Find a chefs knife that feels good in your hand, and take up BDL on his offer, he has put quite a bit of time into the issue. Upon his suggestion I bought "An Edge in the Kitchen" by Chad Ward. The book covers far more than simply how to hold your knife.

I use to be like that, but I've to change that habit, so i practice a lot.