Again, small caveat that to keep this close to ELI25, I am abusing some things (including changes in sign) in a way an expert will notice [they can refer to [1] where the signs and frames are dealt with carefully], but I believe this does not qualitatively affect the explanations you're looking for.

The Brans-Dicke (BD) coupling constant \omega can be constant everywhere in a specifically-modelled universe, and usually is. What varies by point in spacetime is the strength of the scalar field. \omega just relates the strength of that scalar field to its gravitational effects on matter. A BD universe with a higher \omega needs a higher absolute scalar value to deviate from General Relativity (which has no scalar potential field, and no \omega parameter).

One can vary anything in a theory, so a f(\omega)+f(scalar) theory is certainly something you can write down and explore the material consequences of. Usually you get an obviously unphysical theory. Even if the theory is not obviously unphysical, if it forces one to add more parameters that one proceeds to counter-tunes to match the good physical theory without these parameters, what is being gained? Maybe a deeper understanding of the more physical of the theories?

No theory that has an auxiliary gravitational field is properly an extension of General Relativity, it is an alternative to GR. This seems like a fine distinction, but auxiliary fields tend to produce astonishingly different outcomes for matter compared to General Relativity. One has to do headstands to suppress these differences or one gets a pretty different set of orbits (visible in signal-timings between spaceships, or lasers bouncing off the moon, etc) in our solar system, a very different count and/or average shape of galaxies, or a very different "texture" to the cosmic microwave background.

To be a candidate for a physical theory of gravitation, the alternative theory must of course match observations at least as well as General Relativity does. GR is supported by a lot of observations, especially in the weak field limit, and especially where GR differs from Newtonian gravitation far from masses.

The authors show that when they do a headstand to make the effects of the vector auxiliary gravitational field vanish in our solar system but not around neutron stars or small black holes, then matter around the black holes can trigger a gravitational avalanche, making the small black hole bigger than is possible by throwing all the nearby matter into it. In fact, it can run away and grow without bound, with the central mass M exceeding all the matter in the modelled universe.

This is caused by ghosts appearing when one does three things: (1) make the auxiliary scalar or vector field dense around and within these compact massive objects but sparse at a distance, (2) make the coupling of matter to the auxiliary field relevant (rather than vanishingly weak) and (3) add a quantum mechanical matter field. Fluctuations in the quantum matter produce fluctuations in the scalar field, and those fluctuations can take a one-way trip across some notional zero: rather than fluctuating from e.g. + -> 0 -> - and then back again - -> 0 -> +, a ghost gets in the way and keeps the fluctuated mode always negative (which means stronger gravitation than one expects from the local distribution of quantum matter).

> ... very specific technical meaning ...

Ghosts are almost always unphysical -- their presence breaks symmetries of nature that are well-tested, such as the local conservation of the proton mass, or a proton's passive or active gravitational charge. The passive charge is how a proton responds to a large nearby mass, while the active charge is how the proton affects the large nearby mass. One well-tested symmetry is that passive gravitational charge = active gravitational charge = mass.

Breaking the symmetries of the Poincaré group (the symmetries of Special Relativity: invariance under translation, rotation, and boost) with respect to that relationship is usually a sign that one's alternative gravitational theory is unphysical. A proton's mass should be the same on the Earth, the moon, in the Cassini-Huygens probe, in the MESSENGER probe, or in the fusing areas of the sun. You should expect to prove an argument that one can allow the proton mass to differ in the most highly redshifted galaxies, or in cosmic rays ejected from them, or at the formation of the cosmic microwave background (or near neutron stars).

Here it might be instructive to look at a specific example of a different ghost, the https://en.wikipedia.org/wiki/Massive_gravity#The_Boulware%E... which haunts a large family of theories which limit the range of the gravitational interaction to be less than that of the electromagnetic interaction. This adapts the active gravitational charge of the proton, making it weaker at long range. The range-limitation is done by having gravitational waves obey the massive wave equation while light continues to obey the massless wave equation. A second-quantization of these waves produces a massive graviton. The mass is taken to be low, much less than neutrinos, but this still turns out to create problems matching everything that General Relativity does, and at observable large distance scales.

Auxiliary metric fields -- scalar, vector, and tensor -- were brought in to chase away the Boulware-Deser ghost, so that large gravitating structures (which are contain many many protons) match what we see in the sky, while still differing from General Relativity in an area of interest (inflation, accelerated expansion, the apparent MOND relation taken to the relativistic limit, and black hole singularities). But like the old woman who ate a fly, bringing in further mathematical objects to stabilize the relationship between curvature and matter appears to be a losing physical programme.

In the particular family of theories in the headline paper, which descends from this work with massive gravity, the BD ghost may be chased away, but a different ghost associated with an effective mass squared shows up and creates divergences in the field equations around black holes with perfectly reasonable spins, and at both the vacuum-atmosphere interface and shallowly below the surface of a neutron star.

[1] https://arxiv.org/abs/1308.6587v2 published in PRL.