The bridge, used during Saavik's attempt to complete the Kobayashi Maru scenario, was described as an Enterprise-class bridge, which was fashioned after the design of the refit Constitution-class USS Enterprise. (Star Trek II: The Wrath of Khan)
While "Enterprise-class" was a fan fiction designation for the refit design Enterprise vessels seen from Star Trek: The Motion Picture until Star Trek VI: The Undiscovered Country (which was also used behind-the-scenes during the making of the former), the latter film contains a possibly contradictory reference. Blueprints that Montgomery Scott is seen examining are labeled "Constitution-class starship", establishing a name for ships of that design. Since the words "Enterprise class" appear on a sign on a door that is marked "mark IV simulator", its not clear that, if in lieu of referring to an "Enterprise-class starship", this might refer to it being the simulator used by the "Enterprise class of cadets" or that the simulator itself is classified as an "Enterprise class simulator".
Both designations of the starship class names are supported by behind the scenes material and interviews. Andrew Probert in particular mentioned that the design for the refit was originally designated "Enterprise-class", and that name was used behind the scenes. On the introduction of the nomenclature Probert has said, "Richard asked me to start my concepts from Joe Jenning's / Matt Jefferies' Phase II Enterprise but then ours took on a more elegant look under Richard's influence, quickly evolving beyond the original series Enterprise with it's different proportions and lines... a visual upgrade, if you will. It looked like a "totally new ship", to quote Captain Decker, and I was thinking, for a while, that the script might further indicate that it was. To that end, I had tentatively referred to it as an Enterprise-Class ship with the designation of NCC-1800 but that idea dissolved pretty quickly."  He has further elaborated,
"Yeah, I can take the blame for that. In the script, the Enterprise was undergoing a refit, which actually doesn't mean a shape change. It means implementing new technologies and new add-ons essentially to an existing design base. As I developed the Enterprise for The Motion Picture, with Richard, it developed into a totally new design, and therefore I thought, since it was the first of this new starship look, that it should be called Enterprise, little realizing that an Enterprise-class starship had been documented in some previous Star Trek mythology somewhere, because people say, "well, that can't be the Enterprise class because this is the Enterprise class." I don't know. There's always a lot of crossover, you know, misconnecting of fan bases, and things of that nature, but I noticed that people are now referring to it as the refit. I always call it the "A Enterprise," because that's what it ended up being... or as "the movie Enterprise," but yeah,...I'm the one who started that whole mess."At some later point the producers decided it would be more appropriate for the vessel depicted in the Star Trek VI chart designed by Michael Okuda's art department to be a Constitution. For further clarification of this topic, please refer to Constitution-class.