You could be correct as well as I've found references to Cook being associated to the Lady Washington and others that don't mention him at all. Here's a link to the Wikipedia information but, as we all know, Wikipedia isn't a totally fact based source:
https://en.wikipedia.org/wiki/Lady_Washington
Here's a little tidbit on the present ship from a credited site:
n March 7, 1989, the
Lady Washington is launched into the Wishkah River in Aberdeen, as part of the 1989 Washington State Centennial celebration. The vessel is a full-scale replica of the original
Lady Washington, which was built in the British Colony of Massachusetts in the 1750s. Within months, the new
Lady Washington will conduct her first tour of Puget Sound and Columbia River ports.
The original
Lady Washington
was captained by Robert Gray (1755-1806), British explorer and eponym of Gray’s Harbor. In 1987, under the auspices the Gray’s Harbor Historical Seaport Authority, work began to create a full-scale replica of the ship that would also conform to current U. S. Coast Guard safety requirements. Thoroughly researched by historians and constructed by skilled shipwrights, the vessel was launched on March 7, 1989, during Washington’s centennial celebration.
The 112-foot, 99-ton ship soon began touring ports around the state, and was met with cheers in every city she visited. Thousands of maritime and history buffs would come to each waterfront to tour the vessel. The
Lady Washingtoncan carry 48 passengers, and many booked trips aboard the sailing ship.
In 1991, the
Lady Washington sailed on its first trip to Canada, and in 1993 the ship traveled to California, making its movie debut in
Star Trek: Generations. The vessel has appeared in other motion pictures, most notably as the HMS
Interceptor in
Pirates of the Caribbean: The Curse of the Black Pearl.
In 2007, the Washington legislature designated the
Lady Washington as the official ship of the state of Washington. The Office of the Secretary of State for the State of Washington holds a mortgage on the vessel to secure the investment of the people of Washington.