A luxury yacht is named for the chase. Its Dutch ancestor was a hunting word.
Etymonline traces English yacht to early Dutch or Norwegian jaght. That came from Middle Low German jacht, short for jachtschip, literally "ship for chasing." Behind it is the verb jagen, "to chase, hunt," from Old High German jagon. Etymonline also glosses jachtschip as "fast pirate ship."
English met the word early. It appears in the 1550s as yeaghe, meaning a light, fast-sailing ship. Etymonline says it was first a vessel of state for important people. Pleasure trips and racing came later.
That later turn has a royal date. In 1660 Charles II returned to England at the Restoration. The first leg of his journey home, from Breda to Delft, was made in a yacht, according to the maritime history site Maritime Views. Wikipedia says the City of Amsterdam then bought a yacht named Mary and gave it to the king as part of the Dutch Gift. The Mary was 50 feet long, had a complement of 30 mariners, 20 gunners and 20 soldiers, and cost £644.
Maritime Views treats the gift story with some caution. It notes that the story rests largely on Samuel Pepys's diary, and that Pepys knew little about ships in 1660. Wikipedia records that the Mary was the first of some 27 yachts Charles II owned between 1660 and 1685.
The king liked racing. After a year he commissioned a faster boat, the Katherine. In 1661 the diarist John Evelyn recorded a race between the Katherine and another royal vessel, the Anne, "from Greenwich to Gravesend and back." Wikipedia's article on yacht racing says one of them was owned, and sometimes steered, by the king.
The Dutch word for these boats was well known to sailors, but Maritime Views notes it had no link to pleasure sailing yet. After Charles II, the word "instantly became popular." English then built its own vocabulary on it. The verb to yacht appears by 1826, and yacht club, a union of yacht owners usually led by a commodore, by 1834.
The Mary ended badly. In March 1675 she struck rocks off Anglesey in fog and sank, and 35 of the 74 people aboard died.