We seem to be debating over nuances - Representative democracy vs pure democracy.
In a republic, a constitution or charter of rights protects certain inalienable rights that cannot be taken away by the government, even if it has been elected by a majority of voters. In a "pure democracy," the majority is not restrained in this way and can impose its will on the minority.
Most modern nations—including the United States - are democratic republics with a constitution, which can be amended by a popularly elected government. This comparison therefore contrasts the form of government in most countries today with a theoretical construct of a "pure democracy", mainly to highlight the attributes of a republic.
I agree the U.S. is not a pure democracy but rather a kind of democracy.