Would this definition make amphibious assault ships carriers?
Short answerYes; in space, troop-carrying parasite craft can serve the same primary attack-and-defense role as fighters.
NotesThe distinction between a fighter carrier and an amphibious carrier becomes less rigid in space. Landing craft can be the ship's principal way to project force even when their payload is troops rather than missiles or fighter weapons. The same craft can protect the force, move personnel, and support operations away from the parent hull. A vessel organized around those functions therefore still fits the operational definition of a carrier.