Hi Alan.
Here's where the 'simpler' Gravitational fields of Newton and Einstein are revealed in their true nature.
Firstly, you will note that the circular fields represent the Newtonian 4pG fields the all Matter produces.
Secondly, the E^2 fields [the diamond ones] are geometric reflections of the super-positioned E-field components of the same field that Newton modelled with his formula for Gravitation [and Coulomb modelled for Charge interactions - hence their similarities]
Leaving the M-Fields to contribute the last 2pG perturbation fields that affect only objects very close to gravitation Matter bodies [of course all of this is further complicated by the rotations of these same bodies]
ie Newtonian 4pG becomes 8pG closer to Gravitational Matter where objects are influenced by G,E & M fields all at once.
In short - all GEM fields are comprised of 4p CONVERGENT fields 2p INTERACTIVE E-fields and 2p PERTURBATIVE M-fields. [additionally complicating these fields is the fact that G&E fields follow the inverse SQUARE law whilst M fields follow the inverse CUBED law].
Re: the moon interactions wrt to the Earth's on flybys - the E-fields will create an interactive force between each body of Matter in addition to the convergent force created by the gravitation of Matter alone.
The SUN, Earth and moon all produce these 3 quantum level interactions which is normally accounted for in Newtonian & GR math BUT it does not model any interactive forces, only the observed net convergent we term gravitation, thereby ignoring the possibility of any interactive E-field forces at the quantum level.
Accordingly, the moon [when positioned on the same side as the SUN during fly-by] will produce a interactive E-field force on a satellite's charged Matter geometries additional to the Earth's and when it is a Full moon it will create a small force in opposition to the GEM field of the Earth during fly-by.
You can get the same result using the current Newtonian & GE models but they ignore the interactive forces present during equatorial fly-bys and the perturbative forces present in higher inclinations.
In short - by geometrically modelling the 3 fields [and all their quantum] interactions additional forces come into play [ie INTERACTION - opposites attract - similar repel & perturbations] which has an effect on the motion of Matter in the GEM fields of other material bodies.
This is all very 'simple' to explain but much harder to illustrate