I asked about sources out of curiosity, not to undermine what you said.  In a Jamaican context what you say is true - from what I cited (and some other reading) I would say that "Taino" is preferable to Arawak for the Greater Antilles because it is more specific in that context.  Taino is used specifically for the Arawakan people of the northern Caribbean - so it is a subset of the Arawakan people (and thus not interchangeable).  Taino could be considered a single "tribe" within the Arawakan language family.  The Island Caribs were another, as were the Lokono of the mainland.  The Arawakan language family is one of the largest in South America - Brazil, Peru, Bolivia and throughout the Caribbean.  The term Taino is not, to my knowledge, used for groups outside of the northern Caribbean.
The Arawakan people of Hispaniola and Puerto Rico had the most sophisticated cultures, the most powerful chiefdoms.  As far as I can tell, it is to these groups (the "classic Taino") plus their related neighbours (the "eastern Taino" and the "western Taino") that they name Taino applied.  The point is not the word as such, since it was probably not a word that they used for themselves except to proclaim themselves "friendlies" to the Spanish.  There was a distinct culture of the northern Caribbean...before I looked it up I though the Lucayans of the Bahamas were not part of that group, but I was wrong.  Words only work if they can be fit to real meaningful concepts.
The term "Igneri" has been used for the "Island Arawaks" who preceded the Island Caribs in the Lesser Antilles.  This group probably made up the basic substratum of the Island Carib population, and gave rise to the "Kalinago" (or women's language) of the Island Caribs.  The men spoke either a Carib language or a Carib-based pidgin, but apparently it had a lot of Arawak words.
The "traditional" view of the Island Caribs as interlopers has been severely questioned in recent times.  One source separated out three theories of their origin.  The traditional one (and the one I got in school, etc.) was the "Carib replacement" model - that the Caribs invaded and displaced the original Arawak population of the Lesser Antilles, and were spreading northward at the time of Spanish contact.  Another one is the "Arawak continuity" model which says that the Arawaks of the Lesser Antilles acquired a Carib pidgin through trading with the Mainland Caribs.  The third is a reticulate model in which Mainland Caribs moved into the Lesser Antilles and merged with the Arawak population in a model comparable to the Norman conquest of England.  I got the impression that modern scholars leaned that way.  One source cited seventeenth century documents that refer to two groups of "Caribs" in Grenada, the Galini (probably Kaliña/Karinya, or Mainland Caribs) and the Caraïbes (Island Caribs).  In addition, contemporary sources refer to the Taino of the Greater Antilles fleeing Spanish massacres and joining the Island Caribs.  In the Spanish period the Island Caribs changed from being enemies of the Puerto Rican Tainos to being their allies against the Spanish.  (Main source: Raiders and Traders: Caraïbe Social and Political Networks at the Time of European Contact and Colonization in the Eastern Caribbean - Ann Cody Holdren, 1998, Ph.D. dissertation, UCLA; McGinnis's dissertation - cited above - has a lot of quotes from Columbus's journals and from other contemporary sources).
Arawak (or aruaca), as I mentioned, comes from aru which was the Lokono word for cassava flour.  
I am not a historian, just trying to do my best to see what the primary literature says.  The use of the word Taino is standard for the Greater Antilles, but whether it is "accurate" is another matter altogether.  McGinnis points out that Columbus was very selective in how he interpreted data - if you were hostile, then human limbs were proof you were Carib, but if you were friendly, then the heads he found in the village was funerary.  Of course the whole idea of the origin of the Caribs is controvertial.  But the traditional view is by no means accepted without question.  That's what makes history so exciting these days.   23:40, 12 Jan 2005 (UTC)