Conservatives are fond of saying that the word "democracy" does not appear in the U.S. Constitution.
In Article IV, Section 4 the states are guaranteed "a republcan form of government."
A republic (from Latin: res publica, the people's thing) "is a sovereign state or country which is organized with a form of government in which power resides in elected individuals representing the citizen body and government leaders exercise power according to the rule of law. In modern times, the definition of a republic is commonly limited to a government which excludes a monarch."

In my humble opinion, in contemporary times, a "democratic-republic" is a nation where regular elections are held to select the representatives of the people.
In Iceland, yes the prime minister did step down after mass protests but he was replaced by an unelected member of his cabinet and the former Prime Minister is still the leader of the Progressive Party which is ideologically a capitalst, market based center-right party.
Kind of like Nixon and Ford in the U.S., but at least Ford had been elected.

From the very formation of this nation, it is an extremely small group of people (currently 538 of them) called "Electors" who determine who will be president. In 29 states plus the District of Columbia Electors must obey the will of the Electorate but in 21 states, Electors are free to ignore the democratic will of the Electorate