Earlier quoted context omitted.
Okay but there's a difference in the nature of the "pushing down" of a rocket, a plane, and a hovering helicopter, and in how much of an impulse is actually inflicted on an external mass.
Is there?
Now consider a hovercraft. It sits on air. It does not push as much air downward as an airplane of equal weight would push. The amount of air it has to displace is probably proportional to the length of its perimeter (around the sides) multiplied by the height of the cushion of air it sits on multiplied by the air pressure underneath. Double the size of the hovercraft in both horizontal linear dimensions, and you've got double the perimeter, same air pressure (the machine's weight growing proportionally to area) and same cushion height, meaning you're holding four times the weight up with twice the air displacement.
Now consider a low-flying airplane. This is like an inefficient hovercraft, with the ground effect in play. Less air gets "pushed downward" than the same plane flying higher off the ground.
Consider a high-flying airplane. The ground effect is gone, but air's still got viscosity. Fly around the world forever and you'll notice that the amount of air beneath the plane is not actually increasing.
Hovering helicopters have to push more air downward than moving helicopters because they have to fight the stream of downward moving air that they've created for themselves.
If air had virtually no viscosity (suppose atoms were really tiny) then you'd have to fly by somehow forcing air to move downwards. If air had an extremely high amount of viscosity (relative to the mass and power of human-scale mechanical devices) then you wouldn't move it downwards at all -- planes would move around in the air like an amoeba or have a cylindrical conveyor belt surface. Our atmosphere is somewhere between these extremes.