The distinction is way too confusing and the quiz quite humorous but makes the point that a 'ship' is something quite big and likely a commercial venture too boot.
If your sailboat has 1 mast and is Marconi rigged (aka Bermuda or Sloop rigged) you can call it a boat or a sloop or a yacht if you want to take on airs. Even those sailboats with the extra hull are Sloop rigged most of the time.
When speaking to the coast guard by
VHF when requesting a pick up you should refer to your vessel as "Sailing Vessel SuchN'Such" or S/V for short. If you do get picked up by the CG then your boat will become the property of a salvor and it won't matter what you call it.
Some people prefer the more distinctive sounding "Sailing Yacht" or S/Y. I do not think that the CG will be impressed though. In their eyes it either motors or it sails. The confusing part is that most of our boats have engines too and some ships have a sail or two. Grrr.
Another term that loosely describes most of our boats is 'bark' or 'barky', coming from the Spanish/Portugee word barque. If your sailboat has a barbecue on board than using the term barque seems apt unless your barbecue catches your barque on fire. Then it is called a Total Constructive Loss. Go figure.