"we are a secular [Republic]" (we're not a democracy)
Actually, we are a democratic republic. A republic because governing authority is vested in institutions populated by elected representatives of the people as opposed to governing authority being directly vested in individuals, and a democratic republic because those representatives are elected through universal suffrage elections and ultimate political power is vested in the people as a whole not a narrow subset.
In common conversation, in political science literature, in government documents, in treaties, effectively everywhere that matters, the shorthand term for "democratic republic" is "democracy," because it is the democratic nature of that republic that matters, not the fact that it is a republic.
Back when our country was founded republics were rare, most nations were monarchies, wherein governing authority is vested in individuals as opposed to institutions. The counterpoint to "republic" is "monarchy."
Similarly, the counterpoint to a "democracy" in modern parlance is an "autocracy." The Soviet Union, Iran, the People's Republic of China all were or are republics, but not democratic republics, instead autocratic republics.
Living, as we do, in a world where almost every nation is a republic, the important concern is whether that nation is a democratic republic, referred to as a "democracy", or an autocratic republic, referred to as a "dictatorship."
So you can be smarmy about terminology, but the fact is that the usage you are insisting on is not how the word is used in modern colloquial English, or in the academic or professional settings that deal with these matters.
You've got a "I like to insist 'democracy' only be used as it was in the 18th Century" fetish, but the rest of us don't have to accommodate your kink.