The sailors must have started most of the misspellings. They were the nautical types of the 15th and 16th century. They were sailors and were not noted for their literary prowess. The inhabitants that were there were named the Carib by some of his crew. The name given to the local tribe, by the Spanish, was Carib.
Caribe means cannibal.
The Caribs were once thought to have eaten the Arawaks. The Caribs of that time were a warlike tribe of South Americans who conquered the Arawak tribes and displaced them. This happened before the Europeans arrived. Caribbeen or even Caribeen, would seem the best choice for the modern spelling of Caribbean, but it must not have looked as good. The sea that, is to the East of the Carribbean Islands, was named after the people who lived there.