Background information Edit
In the script for "The Wrath of Khan", the USS Reliant was described as an, "'an older, somewhat battered starship of the ENTERPRISE class, with a slightly different configuration". In 1982, this was the designation for ships that would be known as Constitution-class starships after 1987. In that year, in "The Naked Now", the USS Enterprise was identified as a Constitution-class ship. This was further confirmed by a schematic that Montgomery Scott was seen examining in Star Trek VI: The Undiscovered Country where the Enterprise was labeled as a Constitution-class ship.
Both designations of the starship class names were 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 its 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 had 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.
While Probert had claimed responsibility for the designation, the terminology had actually already popped up as early as 1967, when Star Trek: The Original Series was still in production. In his adaptation of TOS: "Balance of Terror", writer James Blish refers to the "Enterprise-class" to describe the similarity of the Romulan Bird-of-Prey to the Enterprise. Author Stephen Whitfield also mentioned the Enterprise-class in his reference book, whereas Producer Robert Justman referred to the Enterprise Starship-class on a memo, dated August 9, 1967, though in both instances the Starship-class was meant, as the Constitution-class was still referred to by the producers at the time. (The Making of Star Trek, pp. 164, 203)
In non-canon publications, the designation was first used as part of FASA's Star Trek: The Role Playing Game to differentiate the newer style ship from the older version Constitution and perpetuated in the later, eventually to apocrypha demoted, reference book Mr. Scott's Guide to the Enterprise as well, and on which Probert incidentally, had served as a technical consultant.