@Logicfest -- Even in the United States, that use seems to be regional. I've never heard the term Jamaica pepper, but I do know what allspice is and I know that shows up in all sorts of food.
Don't get me wrong. We eat a lot of stuff with allspice in it during the Christmas season, but it is one of those spices in heavy rotation throughout the rest of the year, too. It is one of the more popular spices on the planet, in fact.