A republic is a land where political authority isn't formally heritable personal property (e.g., basically a not-monarchy), a republic can also (like the US) be a representative democracy. It can also be an authoritarian dictatorship. It can also be a direct democracy.
The laws bind it.
I suppose since the constitution is completely ignored you might have valid points.
You clearly are using "democracy" in a very peculiar sense. If you narrowly define "democracy" as "lawless mob rule", then your statement is true, but that's not what the word democracy is generally used to mean when discussing political systems.
So yes, it is a personal definition. Scholars don't use this definition. Pundits don't use this definition. The common man doesn't use this definition. It's a BS definition, and if it weren't, it would be pointless and redundant because "majority rule" captures this already.