Cid designed it; Shin-Ra just put up the money for it. As if the fact that his last name is Highwind, along with the fact that he immediately took charge and knew how to operate it as soon as the party acquired it, weren't enough, there's this exchange.

CID: $*#&! First the Airship, then the Rocket, and now, the Tiny Bronco. Shinra took outer space away from me and now you want to take the sky away from me too!?
RUFUS: Oh my... You seem to forget it was because of Shinra, Inc. that you were able to fly in the first place.

He also refers to it as "my airship" once the party acquires it.