It was a popular "fanon" idea in the 90's that Klingons got cloaking devices and Bird-of-Prey ships in exchange for giving the Romulans battle cruisers.
Star Trek XI featured Klingon D7's decloaking during the Kobayashi Maru test in 2258, and surrounding the Nerada a 2233 deleted scene.
Enterprise gave the Suliban cloaking technology from the 27th(-ish) century. It also retconned Romulan cloaking tech from "scary and new" in TOS to "they always had them" in 2152.
Klingon ships being called "Warbirds" was supposedly a mistake in Enterprise, despite their winged design. The term was used again in STXI.
Oh, and there's an old-style Romulan Bird of Prey hidden in the background of the Nerada's giant landing area, along with the Vulcan ship from First Contact.
So: the words "Battle cruiser", "Bird of Prey" and "Warbird" are pretty much interchangable for Klingon and Romulan ships.