Hispaniola was the original colony in the Spanish Main. So, yes, Santo Domingo was the first administrative center.
That being said, by the time the game is set in, Havana had been the central/key Spanish port in the Caribbean for two centuries.
From 1561 onward, all Spanish treasure fleets were to assemble at Havana. This had the effect of concentrating major economic and military resources in the city. By the end of the 18th century she was the third largest city in the Americas, behind Mexico and Lima, ahead of New York and Boston. Moreover, it was also a major naval and merchant ship building port, Santisima Trinidad was built there along with countless warships, treasure galleons and merchantmen.
Point being, Santo Domingo and San Juan were major towns/cities but not the center of military and merchant activity for the Spanish Main. Certainly not from the middle of the 16th century onward. Cartagena and Vera Cruz were the assembling ports for the treasure fleets which then amassed at Havana before being escorted by the Armada back to Spain.