Which is a type of ship that has two masts and the foremast being shorter than the mainmast?

The Bluenose. Canada's favourite and most famous schooner. It's imprinted on the Canadian dime.

I have to say that I thought a ship with two masts was a ketch and one with three masts was a schooner.

Captains Courageous a wonderful movie and where I learned a schooners