Democracy in Colonial America was considered a very bad thing, mobs acting with no restraint and little knowledge and wisdom. Whom you would call a supporter of Democracy would have described himself as a "Republican" or maybe a "Whig" --- later the term replaced Republican and didn't have the bad connotations it started with.
You can somewhat see the development in this: What did Democratic-Republicans believe?
The Democratic-Republicans comprised diverse elements that emphasized local and humanitarian concerns, states' rights, agrarian interests, and democratic procedures. During Jackson's presidency (1829–37) they dropped the Republican label and called themselves simply Democrats or Jacksonian Democrats.