Renaming carriers

[Knjaz]

You can use an aircraft carrier IRL with no planes on it as an anti-aircraft platform and potential refueling point. Doesn't mean it suddenly stops having a massive great runway on it.
Plus carriers can dock multiple fighters not just 1 like the cruiser. My dreadnought has 1 docking module on it, doesn't make it a carrier.

No, you can't.

P.S.: If anybody gets confused by my answer - in short, Carriers do posses short-range AA systems for self-defense, but their AA capabilities are closer to a bunch of guys with Stinger missiles, then to AEGIS cruiser, so to say. Sort of a "last hope" weapon. AA role is being fulfilled by other ships in CBG (Carrier Battle Group), as well as aircraft on board.

Now, there's also a Russian "Admiral Kuznetsov", a mix between missile cruiser and a Carrier, that carries a cruiser-class armament in addition to 40-50 aircrafts (that's a maximum, usually it has less), but it's an exception among his kind, an exception that RuNavy is not going to repeat due to various reasons, main of which is that additional armament takes too much space that could be used for aircraft otherwise, and that role (of missile cruisers) can be filled by other vessels in CBG.
But even Kuznetsov's AA is still short-ranged.