Another paper (alas, poorly written due to ESL issues), by an independent author examines the difference between General Relativity and Newtonian gravity in the case of two massive bodies rather than one massive body and a test particle of negligible mass.
The result, analytically determined, confirms Deur's analysis that the distinction is not immaterial in some circumstances, and that this could be the underlying mechanism of modified Newtonian dynamics.
The metric tensor in the four dimensional flat space-time is represented as the matrix form and then the transformation is performed for successive Lorentz boost. After extending or more generalizations the transformation of metric is derived for the curved space-time, manifested after the synergy of different sources of mass. The transformed metric in linear perturbation interestingly reveals a shift from Newtonian gravity for two or more than two body system.