[readerrrr]

There is a distinction. A gravitational field has a center, so if you are in free-falling towards it you can determine that you are in it.

Take two objects and place them apart to float stationary( the line they make should be orthogonal to the center of the field ). Even though they started completely stationary, they will slowly start to move towards each other, as an unknown force is acting upon them. In reality they are falling towards a common center.

I guess this doesn't work on dimensionless points, so the your comment is correct as far as mathematics are concerned.