I was keeping it simple. The EMB is always below the surface of the earth, in any case, saying that an object orbiting the EMB is an object in orbit around the Earth, perturbed by the moon is just another way of looking at it. For that matter, planets orbit the barycenter of the solar system, which can be more than two solar diameters from the center of the sun.
For earth satellites, GPS for instance, the second harmonic of the earth’s gravity field, J2, is about 127 times more significant that the moon. The J2 perturbation is on the order of 0.00024 times the J0 component, the effect a point mass equal to the mass of the earht, located at the center of the earth. The effect of the moon is only about 0.0000019 times the J2 component, on average.
OK, I was doing fine until the “second harmonic of earth’s gravity field”. I don’t get it and would like to. You clearly have a clue. Care to share?
Too simple, I would say, for a case where an object is "captured" by the earth-moon system, executes a chaotic trajectory within its sphere of influence, and is then "ejected". I don't think the moon can be considered a perturbing influence in such a case, when it is a full partner with earth in the festivities.
Thanks!