I get it that the question itself isn't clear. What I mean is, many people refer to it as La Hispaniola. But if you ask Dominicans, they might say it's Santo Domingo (also the name of their capital) or maybe Quisqueya (supposedly from the native Arawak peoples' name for the island, but probably false). However if you ask Haitians, they might call the entire island Haiti. What's the proper name?
In the end, it doesn't really matter as long as the message is understood. However, the names are a confusing thing of their own. The first name given by Europeans to the island is "la Española", as accredited by Columbus' letter to Luis de Santángel, which I quote and translate.
Del qual cabo avía otra ysla al oriente, distante désta dies e ocho leguas, a la qual puse luego nombre Española y fui allí y seguí la parte del setentrión, asy como de la Juana, al oriente çiento e setenta y ocho grandes leguas por linia rrecta del oriente
Translation: From which cape there was another island towards the Oriente, eighteen leagues away, which I immediately named Española, and went there, and continued on its septentrional side, as well as that of Juana towards the Orient for one hundred and seventy eight leagues on a straight line towards the Orient.
Santo Domingo is a denomination that appeared rather early, in the first decades of the XVI century, as a case of metonymy, the whole being named for the part. The same happened to Puerto Rico and San Juan, which were switched. The island of Puerto Rico was named San Juan, and the main city was Puerto Rico. Eventually, people started calling the island Puerto Rico, the same as with Santo Domingo.
Gonzalo Fernández de Oviedo, who was governor of Santo Domingo, uses both La Española and Santo Domingo in his works. When talking about Francisco de Garay, for example, he says the following:
Antes desto, Francisco de Garay, por su industria e granjerías, fué en esta isla de Sancto Domingo rico hombre e muy aprovechado, e mucho más lo fué después con esta compañía que tuvo en las haciendas reales, de que resultó que estando muy próspero destos bienes que quita y da la fortuna...
Translation: Before this, Francisco de Garay, due to his industry and farming, was in this island of Santo Domingo a rich man, and of profit, and much more he was with the company he had in the royal estates, being very prosperous on these goods that Fortune gives and takes...
The name Haiti is the name the natives had for the island of Hispaniola or Santo Domingo, a fact attested, again, by Fernández de Oviedo. I shall give some quotations on the matter:
De las salinas naturales y artificiales que tenían los indios en esta isla Española, llamada Haití, antes que los cristianos conquistasen estas partes, y de las que hay al presente.
Translation: Of the natural salt pans that the indias had in this island of Hispaniola, named Haiti, before the christians had conquered these parts, and what pans nowadays remain
Pues conviene para conclusión de la primera parte desta General Historia de Indias, dar particular razón de las otras islas, pues he recontado lo que he podido ver y entender de la principal dellas, llamada por los indios Haití e por los cristianos Española...
Translation: For it is convenient to conclude the first part of this General History of the Indies, giving particular report of the other islands, as I have told what I have been able to see and to understand of the main of those, called Haiti by the Indians and Hispaniola by the christians...
A syntagm that appears more than dozen times in Oviedo is "Haiti, que agora llamamos española" (Haiti, that we now call Hispaniola), or any variation thereof. Haiti is not only attested in Oviedo, of course, but in many other chroniclers and travellers, who all insist that this was the name the island had prior to it being baptised as Hispaniola.
Quisqueya is a much less frequent denomination, but Spanish chronicler Francisco López de Gómara, when speaking about Hispaniola, mentions Quisqueia along with Haiti as names for the island. I quote him on this:
EL SITIO DE LA YSLA ESPAÑOLA Y OTRAS PARTICULARIDADES.
En lengua de los naturales de aquella ysla se dize Haití y Quisqueia. Haití quiere dezir aspereza, y Quisqueia, tierra grande. Christóval Colón la nombró Española; agora la llaman muchos Santo Domingo, por la ciudad más principal que ay en ella
Translation: The place of the Hispaniola island, and other particularities. In the language of the natives of that island it is called Haiti and Quisqueia. Haiti means roughness, and Quisqueia big land. Christopher Columbus named it Hispaniola; now many call it Santo Domingo, for the most principal city there is in it.
In conclusion, all three denominations are correct and apply to the same island. Use them at will.