Except that there's dialogue in "Paradise Lost" that contradicts this. Captain Benteen says, "We've been unable to stop the Defiant. Someone has equipped her with ablative armor and neglected to inform Starfleet Command."
Sisko hints at the Defiant's secretive nature in "The Search":
"Offically, she's an escort. Unofficially, the Defiant's a warship. Nothing more, nothing less."
Shouldn't be a big surprise that Starfleet didn't necessarily want word out that they had built a ship to be a straight-up brawler, not to mention it had a cloaking device too.
It's very possible the ship was very top-secret until "Paradise Lost" at which point the beans were spilled. Leyton was not a full admiral and wouldn't necessarily have the highest security clearance.
If the Defiant is unofficial considered a warship, shouldn't the Galaxy class, Nebula, class, Akira class, Prometheus class, and Sovereign class starships be, too?