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 23, Apollo 8 would reach the gentle crest of that hill, the place where the earth's gravitational influence gave way to the moon's.  From then on it would begin falling toward its destination." (Chaiken, p. 95)  All bodies in space have a gravitational field, even an astronaut on an EVA has a gravitational field around them.  In this case, the moon's gravitational influence becomes stronger, and the earth's disappears, even though the moon orbits the earth, it's the moon's 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 8 was the first manned mission to leave the earth's influence.