In TOS, Romulans first had what we call the classic Bird of Prey. It was not called that in the show, the phrase comes from Stiles saying "they had a giant bird of prey painted on them."
Then the Romulnas had Klingon style battlecruisers. They were not referred to as anything except "Klingon ships" on screen.
Likewise when we saw Klingon battlecrusiers in TOS, they were only called "Klingon ships" or battlecruisers. No class name. The term "D-7" comes from an off camera ad lib when Shatner and Nimoy were kidding around. Fandom adopted the designation.
In The Motion Picture we got updated Battlecrusiers. I don't think they were called K'Tingas on screen. Another fan appellation that stuck?
Then in Star Trek III we're introduced to the Klingon Bird of Prey. Now, here's the thing: It was SUPPOSED to be Romulan at first. Commander Kruge was supposed to have stolen a Romulan ship to use the cloaking ability for his mission. But the film dropped the whole Romulan angle for time savings, and they decided to just say it was a Klingon ship from the beginning. Thus began decades of Klingon Birds of Prey, almost on a whim.
During the run of TNG and DS9, we saw many different sizes of BoPs. We heard the larger type called "K'Vort" in Yesterday's Enterprize. I'm not sure where the smaller ones were called "B'Rel" class, if at all. They were probably referred to as Birds of Prey throughout, but the only time I recall hearing it specifcally was in Trek VI.
When TNG decided they needed bigger Romulan ships to threaten the E-D, they devised the giant "Warbirds." These should be the only ships correctly referred to as such.
When Enterprise premiered, we had all this history established for them to draw on. Yet Enterprise had writers whose heads were firmly retracted up their asses and locked. So right off the bat we heard someone refer to Klingon ships as "Warbirds." This was a very careless mistake. Let us just assume Soval was a Vulcan idiot and got it wrong.
The first time we saw a Klingon ship in ENT, even though the art department had been designing a retro battlecruiser, the assholic producers told them to just use the advanced K'Tinga from the movies, 200 years before it was supposed to exist.
After that, ENT introduced a few different good retro Klingon designs, including one that looked like a BoP.
So:
Technically the only thing we should be calling a Warbird is the giant Romulan cruiser from TNG and DS9. The first BoP was the Romulan in Balance of Terror. From then on, all the little Klingon ships were called BoPs. Generally, all the larger Klingon ships are called battlecruisers.