The term "Constitution class" was on a diagram though; I believe it was in one of Scotty's technical manuals.
This is the point where the details matter and become crucial.
"Constitution Class" popped up in the script for "Space Seed" (and apparently along with the primary phaser schematic) belonging to one of the starship manuals Khan had been studying, when McGivers stepped in.
The schematic had a comeback in "The Trouble With Tribbles" on a monitor display in the recreation room but not as a part of a starship manual but as a part of a technical journal.
While this has been the essential and decisive source for the assumption that the Enterprise is a CCS (Constitution Class Starship), it is obviously entirely conjectural.
Bob