These yoga push ups are a multi-part exercises that I developed to teach my students the necessary body awareness for chaturanga dandasana. Turns out that it's a pretty good exercise in its own right.
I'll explain it the way that I normally teach it.
The arm actions for the yoga push up are usually the first thing that I teach. While laying belly down, position your hands so that they are on the floor slightly ahead of your elbows. The first part of the exercise is to push your hands forwards, then down, then relax. Repeat. Because your hands can't move the action of pushing them forwards causes the elbows and shoulders to move back. If you push forwards enough with your hands you may also cause your body to move back.
And that's usually what I show my students if I can see they aren't getting the forward push.
"Push your body back" then relax.
For the next part of this exercise do the same thing but after pressing your arms down, lift your ribs an inch off of the floor.
If you've read balance basics you know what I mean by making a part of your foundation "empty." In this case, push your hands forwards and down to the point that your connection with the floor via your ribs feels empty. Then it's a relativily simple matter to "peel" your ribs off of the floor with hardly any further arm movement.
After you lift your ribs, (but not your pelvis) slowly lower back down and then relax.
To develop your body awareness make that last part into two distinct steps. First touch your body down, then gradually relax.
For part 3, you get to rest your arms.
Instead focus on your feet, knees and pelvis.
Again, develop body awareness by lifting your knees first, then your pelvis, then your belly. And to make these actions even more mindful, try to slowly lift them.
Lower slowly also.
A smoooth continuation from the previous exercise is to bend your elbows and place your hands just ahead of your elbows.
Usually I teach a low lift, just lifting a few inches off of the floor into the yoga pose called chaturanga dandasana. However you could also lift all the way up until your elbow are straight.
Another option is to lift knees, pelvis and chest all at the same time, but smoothly and mindfully.
And lower together also.
When your practice pushing your hands forwards you may notice that your elbows point back as you push forwards. Your shoulders may move back and up at the same time.
As you lift up try to keep the forward pressure in your arms so that your elbows continue to point back.
If you are lifting all the way into an elbows straight position, allow your arms to rotate naturally. You may find that your elbows want to point out. I'd suggest letting them do so.
For extra fun and for improved "tension integration" press your fingers into the floor prior to lifting up. Maintaing finger tension until you lower and relax.
Rest on your belly for a few breaths afterwards to enjoy the after effects of yoga push ups. You might choose to follow with other arm strengthening yoga poses or with a chest stretch like lapasana.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.