How is the External Field Effect in MOND conceptually distinct from Newtonian gravity and GR?

The key to understanding the External Field Effect is through the lens of equivalence principle. How is equivalence principle manifested in galaxy settings?

Let's take a look at an object rotating around a galaxy $G_0$. Its Newtonian acceleration is: $$ a_{Newton} = a_{ext} + frac{GM}{r^2}, $$ where $M$ is the mass of the galaxy in concern, and $a_{ext}$ is the acceleration from the gravity pull of neighboring galaxies. Provided that the neighboring galaxies are not too close, we can assume that $a_{ext}$ is a constant within the galaxy $G_0$.

Now let's hitchhike Einstein's elevator with acceleration $a_{ext}$. In the free-fall (in gravity field $a_{ext}$) reference frame, the observer is moving with the center of the Galaxy $G_0$. In the eye of this free-fall observer, the object rotating around a galaxy $G_0$ is experiencing an acceleration of: $$ a = a_{Newton} - a_{ext}= (a_{ext} + frac{GM}{r^2}) - a_{ext}= frac{GM}{r^2} = a_{Newton}|_{a_{ext} = 0}. $$

The equivalence principle is working: we don't care about the the (uniform) external acceleration, insofar as we move along with the free-fall reference frame. So far so good.

Now turn to MOND. The same object rotating around the outskirt of the galaxy $G_0$ is purportedly in the deep MOND regime, wherein the acceleration experienced by the object is: $$ a_{MOND} = sqrt {a_0 a_{Newton}}= sqrt {a_0 (a_{ext} + frac{GM}{r^2})}, $$ where $a_0$ is the MOND acceleration constant.

Can we hail a ride on Einstein's elevator to cancel out $a_{ext}$? No dice! No matter what you try (try any $a_{ref}$) the acceleration in a free fall reference frame would be: $$ a = a_{MOND} - a_{ref}= sqrt {a_0 (a_{ext} + frac{GM}{r^2})} - a_{ref} neq a_{MOND}|_{a_{ext} = 0}. $$ In other words, you can't get rid of the gravity effect from the neighboring galaxies ($a_{ext}$) . And this violation of equivalence principle is what was observed in the quoted paper as a falsification of the Newtonian dynamics as well as Einstein's general relativity (with or without dark matter), which lends further support to MOND.

OK, to explain how people come up with MOND and why it is different than Newtonian gravity, it might be best to explain where it came from. The core idea is that it is an explanation intended to do away with dark matter. Amongst the puzzles dark matter is intended to explain is galactic rotation curves.

The idea here is that you can look at the velocities of objects rotating around the galaxy, and using Gauss's law:

$$int {vec g}cdot d{vec A} = 4pi G M_{inc}$$,

you can compute $g$ as a function of the distance from the galaxy, and from this, you can compute the velocity of the stars at that radius, using $v^{2}/r = g$.

So, from observing the speed with which stars orbit the center of a galaxy, and using Newtonian laws, you can estimate how much mass is inside a radius $r$.

But, the problem found is that these curves fall off far, far too slowly to be explainable in terms of the observed stars. The common hypothesis to explain this is dark matter, but Milgrom came up with an alternative explantion: maybe gravity needs to be modified.

In particular, he proposed a modification to the law of inertia at low acceleration, that amended Newton's second law to be $F = mu(a/a_{0})ma$, where $mu$ is some function. He then showed that you could pick a plausible value for this function to explain the rotation curves, and that hadn't yet been tested.

OK, so, now this was at least not experimentally ruled out for a while, but it struggled to explain the other evidence for dark matter that was out there. it also did not easily generalize to relativistic regimes. In order to do this, a theory called TeVeS was worked out. This modified general relativity by adding several dynamical fields to the theory (namely, a vector filed and a scalar field to go along with the tensor metric tensor, hence the T, V and S in TeVeS). Then, a Lagrangian could be written down that would reduce to MOND in an appropriate limit and could be used to create relativistic generalizations of MOND to attempt to explain things like the bending of light by globular clusters. This came at the cost of being tremendously complicated as a theory. These are the auxiliary fields that are likely being talked about above.

Finally, just for completeness, I'd be remiss if I didn't point out that most of the active research on this came to a close with the observation of the Bullet Cluster, which was an observation of two colliding galaxies, which produced a clear seperation of the dark matter from ordinary matter, and seemed to provide relatively definitive evidence against explaining dark matter through dynamical arguments (though the complexity of TeVeS left open the idea that the auxiliary fields could be coupling to each other).