(written from a Production point of view)
This story was named "Star Trek III" by Gene Roddenberry himself. This was because he regarded Star Trek: The Original Series as the first iteration of Star Trek, Star Trek: The Motion Picture as the second iteration of Star Trek, and this story was to have been the third version. However, some fans familiar with its premise call it the "Captain Kirk meets Kennedy" script. (Inside Trek: My Secret Life with Star Trek Creator Gene Roddenberry, "Log 21", fn.) In SFX Special Edition 2013 supplement "The A-Z of Star Trek" (p. 106), this narrative was likewise simply referred to as "Kirk Meets JFK".
Immediately after the release of Star Trek: The Motion Picture, a second Star Trek film was ordered by Paramount Pictures. (Inside Trek: My Secret Life with Star Trek Creator Gene Roddenberry, "Log 21") In May 1980, Roddenberry proposed that this story idea be used for that purpose. (Star Trek - Where No One Has Gone Before, p. 71) He submitted it to Paramount in the form of a sixty-page outline. "This wasn't as hokey as it may sound," Susan Sackett commented, about the narrative. "There were sensitive moments, even tender scenes with well-developed characters and well thought out science-fiction concepts. Having given the story his all, Gene optimistically awaited a 'go' from Paramount to begin writing the screenplay [....] He [...] steadfastly believed Paramount would give it a thumbs up. This time he would produce a Star Trek movie his way, not like the last one, a film written by a committee and out of control. He would do it on time and on budget. His hopes were high." (Inside Trek: My Secret Life with Star Trek Creator Gene Roddenberry, "Log 21" & "Log 22")
Following a careful review of the suggested narrative, however, the plot was rejected by Paramount, severely disappointing Gene Roddenberry. "His hopes were dashed that summer [of 1980] [....] The studio executives gave Gene and his script a dishonorable discharge," continued Susan Sackett. "What bothered Gene the most was that they didn't give any tangible reasons for the rejection. 'I think they didn't like time travel,' said Gene, 'which is odd, considering that years later their most successful movie was a time travel story.'" (Inside Trek: My Secret Life with Star Trek Creator Gene Roddenberry, "Log 22")
Paramount's refusal of the narrative was around the same time as they hired Harve Bennett to be instrumental in developing a second Star Trek film, which became Star Trek II: The Wrath of Khan. (Star Trek - Where No One Has Gone Before, pp. 71-72) Even while that film was in development, though, Roddenberry suggested doing his own Kennedy story as the second Star Trek film instead of any plot concepts Bennett thought up. Roddenberry made the same suggestion while the story of Star Trek III: The Search for Spock was being devised, though the alternative plot was somewhat revised by then, with the film's climax now featuring Spock as the shooter on the grassy knoll. "Throughout my tenure on Star Trek," recalled Bennett, "that story came up four times as a substitute for whatever we were planning. Star Trek II, III, IV, V, it didn't matter." (Star Trek Movie Memories, hardback ed., pp. 108, 161-162) According to rumors in the early 1980s, Spock would have had to kill President Kennedy, to set history back on course, by firing a phaser at him in Roddenberry's sequel to Star Trek: The Motion Picture. (Lost Voyages of Trek and The Next Generation, p. 17)
In their reference book Lost Voyages of Trek and The Next Generation (pp. 16 & 17), Mark A. Altman and Edward Gross wrote about Roddenberry's plot involving Kennedy and Klingons as if it hadn't actually existed. They also speculated, however, that rumors of its existence, which had circulated in the early 1980s, may have been based on the second treatment of Star Trek: The Motion Picture, in which Montgomery Scott messed up the timeline and Kennedy helped Kirk put it back as it had been.