The word, yes. Slow cooking meat as a concept, however, is too pervasive and general a concept to be traced to any single time/place (imo).

The specific event (i.e. having a barbecue) is a direct descendant from the Caribbean practice. It actually goes over this in the article, along with a very brief description of how barbecues have been used a political events since the founding of the country.

That etymology seems unlikely since Puerto Rican cuisine has nothing that resembles that style of food.

The Tainos were the natives of not only Puerto Rico, but they're mostly gone now. I have a few relatives who are partly Taino, even my great grandmother was a full native to Puerto Rico. To clarify further the Tainos were not only Puerto Rican natives, they were also native to parts of South America (north east), Cuba, Jamaica, and Hispaniola (Haiti / Dominican Republic). It would be incorrect to attribute it to Puerto Rico itself, thus I attribute it to the Caribbean where they were natives. I'm sure it's been done elsewhere, but it became widespread in the southern states first according to Time:

http://time.com/3957444/barbecue/