The word Dominican can be used when talking about someone from the Dominican Republic OR someone from Dominica. Dominica is a completely separate island located between Puerto Rico and Trinidad and Tobago.
Person 1: I'm Dominican.
Person 2: Can you speak Spanish?
Person 1: No, I'm not 'Dominican' as in Dominican Republic. I'm Dominican as in I'm from the Caribbean island of Dominica.
Someone from the Dominican Republic or someone from Dominica. Dominica is a completely different island.
Person 1: "Where are you from?"
Person 2: "I'm Dominican."
Person 1: "Oh, so can you speak spanish?"
Person 2: "No. I'm not from the Dominica. The Dominican Republic and Dominica are two different islands."