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!