No, it is unfair to call MOND (and at least much of "the like") as a dark matter theory, because it manifestly generates a solution (or solutions) in the absence of any matter at all.

MOND is characterized by an interpolating function between its fundamental constant a_0 and acceleration, such that in the "deep-MOND" regime, F = m \frac{a^2}{a_0} rather than Newton's F = ma.

Both MOND and Newton work in everywhere-vacuum, and in everywhere-vacuum perturbed by a point mass. Neither the vacuum case nor the point mass case qualifies as a "dark matter theory" any more than the vacuum Schwarzschild solution of General Relativity does. These are model universes where one can calculate a Gauss law for gravity.

MOND would behave differently for deep deep extragalactic space with small icy body moving around a large gassy body than Newton would, or than we would get from e.g. a roughly Einstein-de Sitter solution.

Milgrom has pursued several relativistic corrections to the simple MOND formula, and they generally allow for solutions where the stress-energy tensor is everywhere 0, and everywhere 0 except for one point. Milgrom originated MOND. He has collected numerous thoughts and many links at http://www.scholarpedia.org/article/The_MOND_paradigm_of_mod...

When applied to large systems like galaxies and especially galaxy clusters, Milgrom's corrections may be insufficient, and other corrections (catalogued well in https://arxiv.org/abs/1112.3960 section 7). Several of these are very hard to make work with the stress-energy tensor everywhere zero, and those (certainly not all of them) one might ungraciously call "just another dark matter theory". On the other hand, several of these theories admit plausible gravitational radiation in the total absence of matter, just as does General Relativity. (Gravitational radiation is not a feature of Newton or the simple original Milgrom formula.) McGaugh runs tritonstation and is a well known sceptic of particle dark matter.

Relativistic dark matter theories deliberately introduce nonzeros into the stress-energy tensor. You need a distribution of dark matter and trace out how that generates curvature, or you don't have a (relativistic) dark matter theory. You can also of course set out a distribution of cold dark matter and see how it behaves under F = ma or F = m \frac{a^2}{a_0}. Neither MOND nor Newton requires dark matter; likewise neither MOND nor Newton is incompatible with dark matter. Likewise, General Relativity doesn't require dark matter. But one gets more physically realistic solutions at a wide range of scales when one adds dark matter to a defined large-scale distribution of matter and radiation. MOND or the various attempts at generally covariant Milgrom gravitation would also behave differently when one adds dark matter to a large-scale distribution of matter and radiation. That is unlikely to produce more physically realistic solutions to a wide range of scales, however.

Indeed, Milgrom et al's most concrete argument is that their formula for gravitation is worse if one adds in a distribution of dark matter, and that their formula is better than e.g. a Lemaître-Tolman-Bondi solution equipped with a distribution of cold dark matter. That's as much a non-dark-matter theory as I think it is possible to have!