I've never seen the NCC-1701 called anything other than "Constitution class" but I don't know about the NCC-1701-A.
What I am referring to is that the Enterprise from Star Trek: The Original Series may have been so significantly overhauled leading up to Star Trek: The Motion Picture and thereafter so as to be considered a different ship class. For instance, Sternbach, et al, in Star Trek Spaceflight Chronology refers to it as Constitution II class ProfessorTrek 01:07, July 19, 2011 (UTC)
The Star Trek TNG Tech Manual page 3 refers to the TMP Enterprise and the 1701-A as "Constitution class". However, many fans feel that the refit ship should be called "Enterprise class" to denote the fact that very little of the original ship survived the refit. Admiral William Thomas Riker 22:12, July 26, 2011 (UTC)