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Objects accelerate faster in a fall on Earth than they do on the Moon, because each atom of the "falling" object is attracted to each and every atom of the Earth (or the Moon), and the Earth has a lot more atoms.

It's like taking two identical rubber bands and stretching one to twice its unstretched length. That will produce a certain amount of tension. If you then stretch both that same amount, you'll get twice the force.

Same thing for inter-atomic attractions.

Objects accelerate faster in a fall on Earth than they do on the Moon, because each atom of the "falling" object is attracted to each and every atom of the Earth (or the Moon), and the Earth has a lot more atoms.

So you understand that if the Earth has more mass than the moon, that the acceleration of an object towards it is faster, but you don't understand that if the object itself has greater mass, it has the same effect.

The Moon pulls on the Earth and the Earth pulls on the moon. Together, they orbit around spot in space, aleit the Earth is much closer to that spot than the moon. The force holding them is the combined gravity of the moon and of Earth. Objects orbit at their balance point, higher masses require higher orbits or faster orbits. If the entire mass is no different than the individual atoms, why would that be? It wouldn't.