A Ship's Carpenter was a standard part of the complement of sailing ships.
I have a list of the officers and warrant officers of the East Indiaman Duke of York on her maiden voyage (1817 - 1819).
Listed are the Carpenter and first and second Carpenter's mates - all three of whom died at Whampoa between 16 September and 25 October of 1818.
Generally, the term Shipwright implies someone who builds ships (or refits and performs substantial repairs to ships) in a shipyard.
ADDED:
Whampoa also claimed the Chief Mate, the Sailmaker and the Quartermaster, plus at least one of the Seamen.