The Canadair North Star by Larry Milberry defines DC-4M varients as going to Trans-Canada Airlines and the C-4 being the Argonauts of BOAC
I think it's legitimate to apply either designation.
The C-4 was Canadair's design number (the TC also quotes a C-4-1 variant, though I can't see any significant differences between the two, and of course there was also a single C-5).
Confusingly, the DC-4M has a separate TC (two, in fact). The DC-4M was certificated at 73,000 lb MTOW and the DC4M2 (sic) at 80,200 lb.
Trans Canada's aircraft were DC-4M-2 and BOAC's Argonauts, commonly referred to as C-4, were DC-4M-4. The RCAF's unpressurised versions were variously described, confusingly, as DC-4M-1 or C-54GM.