So the legal entities which operate aircraft and are party to labor contracts are carriers.
An airline might be an umbrella company that subcontracts aircraft operations out to carriers. I don't think we've seen a pure subcontract airline in the us (yet). At least not for pax... Amazon.
"Airline" is also obviously a colloquial term for carrier in common usage, since most people in and out of the industry refer to carriers as airlines except maybe regulatory staff.