The existence of the "Starship Class" plaque pretty much excluded the possibility of the Enterprise being the "1st bird of the 17th design". Then again, it isn't the first time the artist's intent is trumped by other production decisions.
Apparently "Starship Class" is just an umbrella term for the most powerful Starfleet vessels and helps to distinct from destroyers and scouts.
However, it wouldn't allow the distinction of inherently differently "Starship Class" designs like the USS Enterprise
compared to the USS Reliant
, a "starship", too.
Consequently, you have a sub-classification, that's proven canon by the (long range) primary phaser schematic seen in "The Trouble With Tribbles".
It's a phaser of a "starship" (= Starship Class) which obviously belongs to the "Constitution Class".
All the ships of the 16th design (Constitution Class), 17th design (Enterprise Class) and of the 18th design (Miranda Class) are "starships".
"Constellation Class" - various novel adaptations (I think James Blish & company)
"Constitution Class" - Bjo Trimble, Greg Jein and Franz Joseph publications
"Enterprise Class" - Making of Star Trek
, Matt Jefferies, official TMP Blueprints