Yes, a republic's head of state is called a president and a King or Queen as head of state is called a constitutional democracy.
It's just all word play. Democracy is where the people make the laws, if everyone voted on all laws, that's pure direct democracy. But everyone can't fit into Congress/the Commons, so you vote for a person to represent you to vote on the laws. Read about Switzerland and direct democracy.
Whether or not the US constitution contains the word democracy, in modern political terminology, the part where you vote for someone to represent you in making laws, that bit is called representative democracy.
Edit - if you say Republic and I say Republic with representative democracy, both are right, it doesn't effect how America runs or is running.