The 367-80 story is really an extension of earlier Boeing designations.
Remember the Stratocruiser? The 377? Its military counterpart, the C-97/KC-97, had the company designation 367. There were several derivatives planned, and one turboprop version (C-97J) went into limited USAF service.
As it became clear that only a turbojet tanker would meet future needs, the 367 tanker/transport series went through several more paper cycles, until the final 367-80 bore no resemblance to the KC-97.
But when the USAF ordered the KC-135, the fuselage diamater grew slightly (4" IIRC) from the -80. The fuselage of the commercial model, the 707, was about 12" bigger than the -80.
And - the 720 was a marketing designation only. The nameplate and type certificate call the "720" a 707-0xx (where the xx is the Boeing customer code).