Before the Europeans realized that the world consisted of more than Europe, Africa, and Asia, the islands off Southeast Asia were known as "The Indies".
Due to the confusion created by Columbus's belief that he was near Cipango when he arrived to the Caribbean (btw, Cipango was the old name for what today is Japan, in fact he actually believed the Cibao to be a local variation of the name Cipango.... my my). In any case, his arrival to the Caribbean lead the islands to be referred as "the Indies" as well, which in Spanish would be "las indias". However, to differentiate the "two" indies in the world, the term West Indies was applied to the islands of the Caribbean and East Indies applied to the tropical islands off the southasia landmass.
Such names of each two regions are Eurocentric terms given that from the European point of view, the Caribbean is towards the west and the pacific islands are towards the east.
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Regarding what Dominicans consider themselves in relation to the rest of the Caribbean islands.... it would have to be Hispanos.
The term "West Indian" was not used to denote native indigenous (Arawaks, Caribs, etc), but rather anyone who was born in the British West Indies. In fact, today the english speaking islands of the Caribbean are still referred to collectively as the British West Indies (also, the French speaking islands (except Haitit) are collectively referred as the French West Indies and there is as well the Netherland West Indies). Notice, however, there is no such thing as the Spanish West Indies. When ever the Spanish island nations are being referred, the term used is "the Spanish Caribbean".
It would make sense that the english speaking islands of the Caribbean would refer themselves as West Indians given the term British West Indies used to denote the english speaking caribbean.
However, there has been (and to a certain extent continues to this day) a distrust between the Spanish and the British. Such distrust spilled into their colonies, the Caribbean being ground zero outside of Europe for the feud to be practiced. The constant British raids on Spanish colonies and the greatest theft of all being the capture of Jamaica by the British and their attempts to capture Cuba and Santo Domingo simply re-inforced the anti-British sentiment that existed in the Spanish Caribbean.
Such distrust exist today, accentuated by the different languages and customs of the British islands vs the Spanish islands that would lead the Spanish islands to accentuate their cultures and identities with the other Spanish speaking nations of the hemisphere.
There also exist a distrust between the Spanish Caribbean and the French, the Dutch, you name it. However, the British were the most dispised.
A good example of this would be Sir Francis Drake. In the British West Indies he is considered to be a hero. Just go and ask a Dominican who knows his history to see what his opinion of Sir Francis Drake is.... it will not be favorable in any sense of the word.
BTW, there is an educational institution in the Caribbean known as the University of the West Indies (or something along those lines). They have campuses exclusively on several English speaking islands. Interesting, no?
-NALs