Cause of weightlessness

The statement in the image is shockingly wrong. Given that the Earth is visible behind the astronaut, the reduction of the gravitational force due to distance must be fairly small; on the ISS for example the force of gravity is about 88% the value at the Earth's surface - very far from being negligible! The reason why the astronaut appears weightless is that she is in freefall, accelerating downwards at $g$.

You are correct that the caption is wrong. The cause of weightlessness is that gravitational and inertial forces cancel, exactly according to the equivalence principle.

Whatever written in the book is wrong. But let us assume it is correct then had it been the case then the moon would have flown away as well as the ISS along with the astronaut.