On purely General Relativistic terms, the Schwarschild radial coordinate is clearly in no sense a distance from the centre in either the Schwarzschild solution with Kruskal-Szekeres extension describing a black hole, nor the Schwarzschild solution matched to a solution of the Tolman-Oppenheimer-Volkoff equations describing a star (R<2M is simply never given any meaning).
My main confusion was why M/R can be given a Newtonian interpretation in the relativistic far-field limit. The Hawking and Ellis reference above is the most painstaking about all the different masses and radii, but two other books with useful discussions and warnings about the usual sleights of hand are:
Schutz, A First Course In General Relativity, CUP 1985
Sections 8.4, 10.5
Hobson, Efstathiou, Lasenby. General relativity: An Introduction For Physicists. CUP, 2006
Sections 9.2, 9.8, 12.4