Interesting! Not what I was taught.
However;
A distinct set of definitions for the word republic evolved in the United States. In common parlance a republic is a state that does not practice direct democracy but rather has a government indirectly controlled by the people. In the rest of the world this is known as representative democracy. This understanding of the term was originally developed by James Madison, and notably employed in Federalist Paper No. 10. This meaning was widely adopted early in the history of the United States, including in Noah Webster's dictionary of 1828. It was a novel meaning to the term, representative democracy was not an idea mentioned by Machiavelli and did not exist in the classical republics.[50]
The term republic does not appear in the Declaration of Independence, but does appear in Article IV of the Constitution which "guarantee[s] to every State in this Union a Republican form of Government." What exactly the writers of the constitution felt this should mean is uncertain. The Supreme Court, in Luther v. Borden (1849), declared that the definition of republic was a "political question" in which it would not intervene. In two later cases, it did establish a basic definition. In United States v. Cruikshank (1875), the court ruled that the "equal rights of citizens" were inherent to the idea of republic. The opinion of the court from In re Duncan[51] (1891) held that the "right of the people to choose their government" is also part of the definition. Due to the 1875 and 1891 court decisions establishing basic definition, in the first version (1892) of the Pledge of Allegiance, which included the word republic, and like Article IV which refers to a Republican form of government, the basic definition of republic is implied and continues to do so in all subsequent versions, including the present edition, by virtue of its consistent inclusion.
Beyond these basic definitions the word republic has a number of other connotations. W. Paul Adams observes that republic is most often used in the United States as a synonym for state or government, but with more positive connotations than either of those terms.[52]
With no monarch, most modern republics use the title president for the head of state. Originally used to refer to the presiding officer of a committee or governing body in Great Britain the usage was also applied to political leaders, including the leaders of some of the Thirteen Colonies (originally Virginia in 1608); in full, the "President of the Council."[40] The first republic to adopt the title was the United States of America. Keeping its usage as the head of a committee the President of the Continental Congress was the leader of the original congress. When the new constitution was written the title of President of the United States was conferred on the head of the new executive branch. Today almost all republics use the title president for the head of state.
If the head of state of a republic is also the head of government, this is called a presidential system. There are a number of forms of presidential government. A full-presidential system has a president with substantial authority and a central political role. The United States was the first example of such a system, and the basis for the model adopted elsewhere. In other states the legislature is dominant and the president's role is almost purely ceremonial and apolitical, such as in Germany and India.
These states are parliamentary republics and operate similarly to constitutional monarchies with parliamentary systems where the power of the monarch is also greatly circumscribed. In parliamentary systems the head of government, most often titled prime minister, exercises the most real political power. Semi-presidential systems have a president as an active head of state, but also have a head of government with important powers.
The rules for appointing the president and the leader of the government, in some republics permit the appointment of a president and a prime minister who have opposing political convictions: in France, when the members of the ruling cabinet and the president come from opposing political factions, this situation is called cohabitation.
Also very interesting!