You have it wrong as well. A Constitutional Republic has to have a Constitution, a representative republic doesn't.
Thomas Jefferson, Resolutions Relative to the Alien and Sedition Acts
The Constitution is the primary driver as it establishes rule by law.
Democracy vs Constitutional Republic - What is America's Real Form of Government
OK, I’ll respect your knowledge on the subject.
So, if someone on the street says, “What kind of government do we have?” what would you say?