That people think of this as a "mistake" is akin to thinking that all references to Starfleet Command must be mistakes, because Kirk's boss was firmly established to be Space Central originally...
Or actually it's even sillier than that, because nothing was "originally" established about how Klingons designate their ships. There is no "Space Central" or "UESPA" to compete with the later Starfleet Command and Starfleet there: Klingons have always had Birds of Prey (which the Romulans have never had, really, as per dialogue, pay heed!), and Klingons have always had Raptors, and Klingons have always had Warbirds.
It may be a coincidence that both Klingons and Romulans like bird terminology for flying things. Or then it's the result of the apparent fact that Vulcans were the ones to first establish this terminology, to translate the native terms to English. It would then suffice that Vulcans are fond of bird terminology (a natural assumption, as they are culturally related to Romulans who have a big bird fetish going for their national emblem and all), and the native Klingon ship designations actually refer to crawling beasts, or are dry technical terms.
Timo Saloniemi
In TOS, we were told... unambiguously... that the one Klingon ship design we ever saw was a "Klingon Battle Cruiser." And we were told that the only Romulan ships we ever saw were "Romulan ships," except for the one time we saw them flying "Klingon designs."
The description of the Romulan ship as "Bird of Prey" comes from the line by Styles about the ships being recognizable because they're painted as "Birds of Prey." From that point onwards, it was fan-accepted that these ships were "Romulan Birds of Prey" but this was never officially stated (well... I'm not recalling every line from "The Deadly Years" however, so it's possible I might be overlooking one from there?)
Now, in the years between the end of the series and the first flick, everything was fandom-based. The use of the terms "Klingon Battlecruiser" and "Romulan Bird-of-Prey" became totally accepted during this time (including being the officially-sanctioned names given to the AMT kits).
During this time, a non-licensed Trek game came out, made by Lou Zocchi, called the "Starfleet Battle Manual." This predates "Starfleet Battles" and, honestly, I prefer it to SFB, though it's far less deep. Zocchi arranged with a miniature-and-dice-maker, Gamescience, to make starship miniatures. They licenced that with Franz Joseph but not with Paramount.
So, you had a full set of Federation ships... Dreadnought, Cruiser, Scout, Destroyer, and Tug... and you had a choice of a "K-type Cruiser," an "R-type Warbird," or a "T-type Webspinner." It was clear what they were (and the models and graphics were near-perfect) but they didn't use any of the copyrighted terms.
This is the FIRST TIME that the term "Warbird" was used in the description of Romulan ships, but it caught on. I believe that when SFB came along, they originally used the Gamescience ships as well, so they adopted the same nomenclature.
Now, for ST-III, the idea was that it was a Romulan Bird-of-Prey. Some descriptions of drafts I've read talk about the villains being Romulans, but that's not something I've seen confirmed. I did read a synopsis with an additional subplot involving Klingons stealing a Romulan ship, however... and that scene got cut while the script never caught up to the change. So ST-III ended up talking about a "Klingon Bird-of-Prey."
At the time, the fans were in an uproar over this "misstatement." But, since it was a new class of ship, it was accepted that ... well... there's no reason that both Klingons and Romulans can't have their own ships called "Birds of Prey," is there?
When TNG came along, they officially called the Romulan ships "Warbirds" (though this was a new class as well).
SO... I have no real problem with the idea of various ships being called various names. As long as they're not supposed to be "the same ship."
But, all things being equal, I'd far prefer Klingon ships to have some other sort of name... perhaps "Hounds" or so forth... implying hunting packs. It seems more appropriate for who the Klingons are. I just don't see Klingons as being all that similar to falcons and hawks and the like... I see them as being more like wolves.
Ultimately, I suspect that the nomenclature that's being used is the Federation's "code names" for the ships anyway. It's like how we call Eastern helicopter designs names like "HIND" and "HIP" and so forth... those are terms WE came up with, not terms that those who make and use the aircraft came up with.
