If details matter to you, then you would know that "carbine" has become a rather loose term. It originally referred to shorter muskets, and that was that. Nowadays there are so many variants of pistols, rifles and anything in between, that the line has blurred significantly. In part due to marketing.
Yes, carbines are for the most part recognized as compact rifles. I.e. any rifle with a barrel lenght under 16 inches (shotguns not included). But you also have designated pistol-caliber carbines (PCCs), which already add to the confusion. It's the manufacturers/marketeers who opted to use the term "carbine" or "pistol-carbine" when referring to removeable stocks and conversion kits for pistols. This goes way back, like with the Mauser for example:
View attachment 527408 Even modern day PDW conversion kits for glocks etc still use the term carbine on various occasions:
View attachment 527412View attachment 527410 Lastly, you can't always apply firearm logic to airguns. There is no such thing as "rifle ammo" and "pistol ammo" in the airgun world. A .22 air pistol will shoot the exact same pellets and slugs as a .22 air rifle. So I hope you can see why people use the term rather loosely here. Definitions change/evolve over time, that's how language works.