When Columbus arrived at the island of Hispaniola, did he plant a cross and say "this is India and I am taking possession in the the name of...

More or less, yes, although I doubt he stated "This is India". There was actually a fairly standard protocol that the conquistadors followed whenever they made contact with a New World civilization. Generally speaking, the indigenous population greeted the arriving ships with open arms, as they had no idea what the motives of the Spanish explorers and conquistadors were, which was to claim not only the land, but everyone on it and their souls for Spain and the Catholic church.

The protocol went something like this: First, the ships would be spotted by the tribe(s) in question, and many of them would congregate on the shore anticipating their arrival. They would greet the Spanish with a big party, offering friendship, feasting, etc. In turn, the Spanish would engage in a show of force to demonstrate their technological and military might, often by aiming their ship's cannons at a tree or other object on the shore and blasting it to bits. This was meant to instill awe and fear. Then, indeed they would at some point plant a staff in the soil and declare that everything the staff touched, and all that was on the land it touched (including the people) was now the property of the Spanish crown and subject to Spanish authority, which also meant the people were all Catholics and subject to all the requirements of becoming and being Catholic. It did not matter that the indigenous audience did not understand a word of it, or even in the cases where they had translators available, grasp the implications of it.

I should point out that the greeting of the European explorers with open arms routine generally applied only to their first contact with them, since it didn't take the Native people long to figure out what the Europeans really had in mind. Also, it should be pointed out that following first contacts, after the Spanish or other Europeans came and left, anywhere from about 25% to 90% of the population soon succumbed to diseases like small pox that the Europeans carried to which the Natives had no immunity. This was of course an unintended consequence, but it did make subsequent efforts to subdue the Native population that much easier.

My sources for this are my degree in anthropology and minor in Native American Studies, which covered all this pretty thoroughly.