The only HMS Quaker was a purchased ketch of 78tons builders' measurement, so we can discount her.

A crew of 550 would be about right for a third rate 64 gun ship or an undermanned 74 gun ship. Most RN ships of this period were undermanned, as there were very many ships in service in time of war.

Sea battles between two or three ships weren't called 'the Battle of Something'. They were called actions (usually single-ship actions) and were termed Shannon and Chesapeake or similar.