The one-arm push-up seems like the logical winner. It provides the opportunity to grease the groove. It addresses a whole lot of symmetry and reflexive stability issues, which is one reason Gray Cook is a fan of The Naked Warrior. It teaches high tension skills. In addition to working the “pressing muscles,” it devastates the midsection.
The last item is an asset and a liability. When your pressing muscles get strong enough for you to knock off multiples sets of five, one-arm push-ups beat up your your abbies, glutes, and QL a lot more than your triceps. Which is why the one-arm push-up does not win this contest.
Note: If your choice of a press has to be body weight, it is a good idea to rotate one-arm push-ups and handstand push-ups every two weeks.
You want some neck extension - look in the same direction as you were when you were standing, i.e., if you were facing a wall before you bent forward into your handstand, don't keep your neck neutral and look at what would be behind you if you were standing, rather look at the wall, at least somewhat.
I cannot explain why to you, except to say that this is what everyone who does these successfully does, or at least almost everyone.
The picture on the left is not usual, in my experience; the picture on the right is what seems to work best for most.
@Karen Smith
@Rif