This split between "democracy" and "constitutional republic" is something being pushed more and more, but it's really a fantasy peddled by some US-based politicos with vested interests in devaluing the concept of democracy and making you accept the fact that you should be happy without it.
A "constitutional republic" is something defined by a set of laws with certain specific elements in common: having a constitution and being a republic (which is also quite a loose term, used to define "anything that is not a monarchy" -- Iran is also a constitutional republic, for example, albeit a theocratic one).
A "democracy" is not a comparable entity. Can you please point out any country who defines itself, in official terms, as a "Democracy"? Even classic Athens didn't call itself such a thing. This is because democracy is not a set of laws: it's a declaration of principles, an abstract utopia of where your laws should come from (i.e. "the people"), traditionally associated with some sort of voting system. On the formal level, there is no clear definition of a complete form of government called a "democracy"; it's commonly assumed such a thing involves some sort of voting rights (better if universal), but that's about it. The term is used to indicate the utopic state where government is carried out "by the people" in its totality. There has been no such perfect government throughout history and probably there never will be, but it's an utopia from which real forms of government can find justification and inspiration.
You can compare a "constitutional republic" to a "constitutional monarchy" or an "absolute monarchy": they are all specific forms of government with very specific common formal traits. A "democracy" is a completely different concept. You live in a constitutional republic based on democratic principles, i.e. a democracy... albeit a flawed one, like all of them are. You should strive to make your republic as genuinely democratic as possible, not renounce your rights "because after all we've never been a democracy".