Heavier objects do "fall" faster

Ok so let’s say the universe is comprised of 2 objects at any given time. say the moon, and another copy of the moon. Alternatively the 2 objects could be 1 moon and 1 ping pong ball. I’m having trouble understanding this because it just doesn’t make intuitive sense that 2 objects of great mass wouldn’t be drawn to eachother at a faster rate than 1 massive object and one object lacking mass. The way I’ve always heard gravity described is as like a bed and you put heavy stuff on it and it causes depressions. And it seems to make sense given that illustration that 2 heavier objects would more easily be drawn to eachother from a greater distance and greater acceleration.

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