1kg mass and a 2kg mass do not fall at the same rate. The Gravitational force is (G*m1*m2)/r^2. You are observing that m1 (the earth) is much much greater than m2 (the 1 or 2 kg masses), and you are simplifying to (G*m1)/d^2 because of the precision of the measuring device. Also, d is the same for both masses.
The force is on both objects at the same time. The force in F = ma is a function of the mass of both and their distance. If the mass is different in the two scenarios, then the force is different. On earth with small weights, they seem the same because of the precision of the measurement.
Is what you're getting at the fact that the distance between the earth and the other object changes from two effects (the first being the ball falling towards the Earth and the second being the Earth falling towards the ball)? That's right, of course. But that distance's second derivative is not the acceleration a in F=ma. Indeed, in both Galilean and Einsteinian relativity acceleration is detectable locally without a needed reference to another object.
Yes - I was making a mistake. I was trying to describe the effect of both masses. When one is much smaller than the other, then the movement is mostly in one direction. When they are closer in mass or even equal, they move toward each other. For example, if you have a 1 liter water bottle filled with a material that gives it the same mass as the earth, then the two bodies will move toward each other, and the water bottle will seem to move toward the earth much faster that the 1 filled with water (1kg). If it is filled with a material, that gives it much grater mass than the earth, the earth will move toward it.
The mass of the earth dictates the acceleration of the individual masses towards the earth. However the acceleration of the earth itself towards the masses are dependent on how much mass is falling towards the earth. When more mass is falling to the earth, the earth accelerates towards the masses faster. So the thought experiment is flawed because with only one 1 kg weight falling towards earth, the gap between the weight closes slower than when there are three 1 kg weights spaced 1 m apart and dropped simultaneously.
If you define fall as the size of the gap. You could also take it as acceleration towards the barycenter, which would be the same. These are indistinguishable for everyday objects so could argue that the word “fall” could be interpreted either way.
