Fifteenth-century nautical terminology is a morass, and the surviving information about the three ships of Columbus is almost non-existent. It seems to be generally agreed, though, that the Nina and Pinta were of the vessel type known as the "caravel." (In one of his journal entries Columbus refers to them as "the two caravels.") He calls the Santa Maria a "nao," which is just Spanish for "ship." Some quite knowledgeable people have suggested that she was indeed a "carrack."
What people of the time actually called these vessels probably depends to some extent on the definitions of the various words - which apparently were pretty sloppy. My usual approach to such problems is to look them up in the relevant volume of Conway's History of the Ship, which is the most up-to-date, comprehensive, scholarly treatment of the subject I know. But its authors wouldn't, I'm sure, go so far as to say their use of terminology for such early periods is really definitive. For what it's worth, here are the terms as defined in the glossary of the relevant volume, Cogs, Caravels, and Galleys: The Sailing Ship, 1000-1650:
"Caravel, carabela, caravela, caravelle. Relatively fine-lined Portuguese craft of the fifteenth and sixteenth century, originally a fishing craft or coastal trader, but most famously associated with the great voyages of exploration. They were originally lateen rigged on two or more masts, and were known for their weatherly qualities, but later variants adopted square canvas for better performance befoe the wind.
"Carrack. The derivation of the word is uncertain: there were small Arab karaques in the thirteenth century, and the term may have been passed to the West via Muslim influence in Iberia, but the ship type seems to owe nothing to the Arab craft. The carrack was a development of the northern European cog...combined with some featues of the local skeleton-first and multi-masted traditions. There are Venetian references to such vessels from 1302-1312....In English documents the words carrack or tarit occur from 1350, applied to vessels of this type and usually Genoese in origin. The carrack seems to have acquired more sail from quite early, a Catalan contract of 1353 specifying main and mizzen, and the English captured a nuber of two-masters from the Genoese early in the fifteenth century (one was actually renamed Le Carake). By the middle of the 1400s three-masted examples were known and the multi-decked forestage and aftercastle were becming more marked. Carracks tended to be very large for their day, and with the application of the hul-mounted gunport, carracks became the capital ships of sixteenth century neavies, until supersded in the latter half of the century by the galleon."
Note that this impressively erudite discourse never quite gets around to telling us just what the word "carrack" means. That's probably intentional. The terms was applied so sloppily, over such a long period, that it's tough to define for the modern reader.
"Nao. One of many terms ultimately derived from navis, the Latin for 'ship'; for most of the Middle Ages in the Mediterranean it wa applied to the most common form of sail-powered merchant ship, only relatively new or unusual typaes being distinguished by individual names like cocha or caravel. Therefore its characteristics varied, and it was used of both carracks and smaller craft in some contexts; in fifteenth-century Catalonia, for example, nao implied a vessel of one or two deck[s] ranging in size from 300 to 700 botti. Columbus's famous Santa Maria is usually described as a nao of this type." ("Botti" was a measure of cargo capacity, based on a barrel called a "botte.")
Bottom line: Columbus himself clearly called the Nina and Pinta "caravels" and the Santa Maria is generally agreed to have been a "nao," but the latter term is so vague as to cover almost anything. It's entirely possible that the Santa Maria met somebody's contemporary definition of the word "carrack."
This stuff is, if nothing else, a great cure for insomnia.