Then there is the dedication plaque on the bridge that simply reads "Starship Class."
To visit the "Constitution Class" TOS Enterprise, please step through the quantum mirror over there and search for the TNG-universe or slingshot around a star and head forward 80 years. Watch out for the Borg-conquered universe with the crazed Riker. He's not too friendly.
I jest. But yeah, the time when actual on-screen evidence directly points to the TOS Enterprise as "Constitution Class" is in a later series like TNG's "The Naked Now", "Relics", etc. From the TOS time-frame, she's got that plaque that designates her as "Starship Class". Then in TWOK, she's "Enterprise Class" and finally in TUC, the -A is a "Constitution Class". It's a bit like "The Naked Now" where history talks about Captain Kirk and we know he was an Admiral at one point. It just depends on where and when you ask what class the TOS Enterprise is