Democracy is inherent and implicit in a republic. And I think you're mistaken about "constitutional republic." That term really wasn't used much, though representative democracy and "democratic republic" were terms that were used more frequently.
The entire argument about the US not having a democracy but a republic really stems (in modern times) from the John Birch Society.
The entire argument about the US not having a democracy but a republic really stems (in modern times) from the John Birch Society.