Bottom line, the conflict between the "Enterprise Class" and "Constitution Class" references in the movies is simply a mistake.
Not a mistake, just a misunderstanding.
Bottom line, all the three decisive TOS creators / producers had settled for the TOS Enterprise
to be a member of the "Enterprise Class", the evidence is Matt Jefferies pre-production sketch (and later interviews) according to which the Enterprise was "the first bird" and the "first of the series", Bob Justman's reference to "Enterprise Starship Class" and Roddenberry's / Whitfield's "Enterprise Class Starships" in The Making of Star Trek
Whether Roddenberry's reference to "Constitution Class" starships in the TMP novelization refers to the TMP Enterprise
is inconclusive. On the other hand, the Official TMP Blueprints
, approved by Gene Roddenberry, refer to the TMP Enterprise
as the "new Enterprise Class".
You correctly observed yourself the internal differences between NCC-1701 (Enterprise Class) and NCC-1701-A (Constitution Class) and therefore provided a clue to solve the riddle, IMHO.