Well you almost have it right but not quite Its like going over a hill  To quote Chaiken It was as if the moonship were coasting up a hill one that became less and less steep as it went along  About two days from now on the afternoon of December  Apollo  would reach the gentle crest of that hill the place where the earths gravitational influence gave way to the moons  From then on it would begin falling toward its destination Chaiken p   All bodies in space have a gravitational field even an astronaut on an EVA has a gravitational field around them  In this case the moons gravitational influence becomes stronger and the earths disappears even though the moon orbits the earth its the moons gravity that is having an effect on the spacecraft when it reaches a certain point in space  Therefore it is correct to say that Apollo  was the first manned mission to leave the earths influence