I'm quoting what you wrote.
Robert Comsol wrote:
It's a phaser of a "starship" (= Starship Class) which obviously belongs to the "Constitution Class".
You wrote, "Starship Class which obviously belongs to the Constitution Class." If the Enterprise is a Starship Class, by the logic in your writing, she's a "Constitution Class".
And to recap, if the Enterprise was either "Starship Class" or "Constitution Class" she cannot be the "1st bird of the 17th design" as she is not the lead ship.