And the role most Star Destroyers play on the battlefield is a hybrid of the roles of battleship and carrier.
Battleship/Carrier/Assault Transport actually. The key point was that a Star Destroyer is supposed to be able to independently cover all roles (excepting picket/skirmish duty). It's a jack-of-all-trades warship, and it does a really good job at all three of its assigned roles. For almost any problem, the question isn't whether a Star Destroyer is the right ship, it's how many are needed based on the scale of the job.
True but I still don't see what the big deal about one character feeling it is a cruiser, and a group feeling it is a battleship is a big deal. The classification says that in the short form the term battleship was used for ships whose mere presence could have a major influence on the battle. I would say an ISD qualifies.