What is the origin of pressure of gas acting a surface beneath it?

I understand that the origin of atmospheric pressure is the weight of air.

If we look to our atmosphere as a container full of gas molecules, each of them have kinetic energy and gravitational potential energy. So, when we talk of atmospheric pressure as a result of the weight of the air above us, we are taking in consideration that potential energy.

An example of that potential energy is the pressure differences even on small $Delta h$, responsible to keep baloons floating.

Does the pressure of gas inside upper compartment come from the weight of the gas (like the case of atmospheric pressure) or the bombardments of gas molecules onto the piston?

In the case of the OP example, only the kinetic approximation for ideal gases are normally used because the potential component (weight of the gas) is very small.

So the pressure results from the collisions of molecules with the container walls. It depends on the number of collisions per time and of the average change of momentum of each collision. What is a function of gas density and temperature:

$$P = frac{F}{A} = frac{MDelta v}{ADelta t} = frac{nmDelta v}{ADelta t} = frac{n}{Delta t}frac{mDelta v}{A} = frac{n}{Delta t}frac{Delta p}{A}$$

Of course, the total pressure on the lower chamber is: $P_t = P_u + W_p$ where $P_u$ is the pressure in the upper chamber and $W_p$ is the weight of the piston.

Normally the air pressure is far more important than the piston weight.

A piston of steel with an area of $A = 0.01m^2$ and a thickness of $t = 0.01m$ weights $7.7N$. If the lower chamber has atmospheric pressure, there is an upward force of $10^5 * 0,01 = 1000N$

If there is no air at the upper chamber, the piston weight is no match for the upward force.

I think your confusion might be coming from your idea that "both the collision of gas with the wall and the weight of the gas result in a downward force". In practice you only need to consider the force due to the collision of gas molecules (aka the kinetic theory of gas).

In your thought experiment, so long as you know the gas pressure (P) immediately above the piston (of area A), you can immediately calculate the force on the piston as P*A. This is independent of the total mass of air above the piston. So if the upper chamber of the cylinder is sealed at the top then you can add or remove gas to/from the upper chamber and the force on the piston will scale exactly with pressure - independent of the volume of the upper chamber and the mass of gas in it.

If the upper chamber is open to the air then the pressure on the piston is then also proportional to the the mass of air in the column above the piston. However, in that case you should only consider either the pressure at the piston top or the mass of air - not both.