Caribbean vs Jamaican

Am I overly patriotic because I am uncomfortable when I see someone who is Jamaican referring to themselves as “Caribbean-born” or “migrated from the Caribbean”? I know to some using the term Caribbean sounds exotic and inclusive, but it seems to me like hiding of their heritage. I remember a time in South Florida when Jamaicans has a bad repetition because of gangs but we would still say “I are Jamaican”. On the other hand I remember when the vicious rumor about AIDS originating in Haiti was circulated  I knew some Haitians who would say they were “French” or even “Jamaican” so as not to deal with the stereotypes.