The Enterprise is identified as a star cruiser, like the Valiant, in the episode "A Taste of Armageddon".
What writers identified the Enterprise as a Constellation-class starship?
Well, the memo from Bob Justman of August 9, 1967 only indicated the names he liked from D.C. Fontana's proposed list. I don't think it was meant to be the exhaustive list. For example, it doesn't have the Defiant which was still a year from being created, but his list also didn't include the Exeter--which was among D.C. Fontana's suggestions. In the end, it looks like the list of Starships was settled upon by Roddenberry regardless of what Bob Justman's preferences might have been.
Of course, the "Constitution class" script comment first appeared in the "Space Seed" script by Carey Wilbur and Gene L. Coon. It's there not only in the Second Revised Final Draft script dated December 13, 1966, but also in the Final Draft script dated a week before that, if not in earlier drafts.