TNG pretty much beat us over the head with the term "warbird" being exclusive to Romulan ships
...And TOS beat us over the head with the term "starship" being exclusive to the class of ships Kirk's Enterprise
was a representative of.
Also, why can't there be "Warbirds" and "Birds of Prey" in both Empires? Could be they are just translations of Romulan/Klingon words into Federation Standard.
Or even Vulcan terminology for starships, considering that we explicitly heard Vulcans providing our Earthling heroes with the "warbird" concept in the pilot episode, and that Vulcans would be the likeliest source for terminology relating to the Romulan war machine just prior to the big war...
We know Romulans are big on bird themes. But they supposedly were that even before the big rift, as they were known as "those who marched under the raptor's wings"... It may well be an old Vulcan thing to regard spacecraft as avians.
Regarding the D-7 designation, we might note that the term D-5 was used in an exclusively Klingon discussion (supposedly conducted in Klingonese) in "Once More Unto the Breach", making it a bit less likely that this would be a "NATO name" given to the Klingon designs by their enemies.
As for it being an overarching designation for a family of designs, it's a good idea as such - but D-12 in ST:GEN seemed to be a very specific designation for a narrow sub-type of Bird of Prey. Unless we read Worf's comments there as indicating that this is "an old type of D-12 BoP" rather than "D-12, an old type of BoP"...