Did the Founding Fathers of the USA really make a distinction between a Republic and a Democracy (and despised the latter)?

Question:
If the USA really was meant to be a Republic, then can someone provide the moment at which Democracy became celebrated as something very desirable in the US culture?

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I'm going to answer the second part of this question first. The people of the United States began to become enamored with the idea that the country should be a Democracy rather than a Republic with the 1824 election.

The first U.S. presidential election in which the people had any direct means for registering their support for a candidate was the 1824 Election. This was the first time popular vote was taken, even though (as today) the popular vote was not binding. This election was also the first time a candidate who won the most electoral votes, states, and the popular vote lost the election. Andrew Jackson won the plurality of the electoral votes, but running against three other candidates he failed to win the required majority. The Constitution left the decision up to the House of Representatives, who chose John Quincy Adams over Andrew Jackson in a back-room deal involving promises of office between Adams and Henry Clay, who was the third of four candidates.

(Another interesting fact about this election is that both Adams and Jackson chose the same Vice Presidential candidate, John C. Calhoun, who ran unopposed for the slot.)

Henry Clay was not just another candidate, but was also Speaker of the House and played an important role in deciding the "election" once it went to the House. Clay bartered with John Quincy Adams to give him the Presidency in exchange for naming Clay his Secretary of State. The office of Secretary of State was an important stepping-stone to the Presidency at the time, as all three of the previous presidents - Jefferson, Madison, and Monroe — had first served as Secretary of State. Only George Washington and Adams had bypassed this office, which had not existed before their terms.

Jackson, who was a war hero with immense popularity, attacked the deal as corrupt. He and his followers founded the Democratic Party on the principle of reasserting the people's will. However, the people never had direct control of the government, which was actually set up as a constitutional republic. Jackson nevertheless would use John Quincy Adam's term and his own two terms in office to champion popular democratic rule and lambast the concept of a republic as anti-democratic, anti-representative, and fundamentally corrupt. Not just because it had robbed Jackson of what he believed should have been his first term, but also because the practice of politicians placing themselves above the will of the people was a recurring irritant for Jackson, an outsider elevated by his war record rather than political achievements. Jackson had the personal prestige to make this point and drive it home.

Some of Jackson's legacies as President are related to democratizing the country. The first truly public national election for president was the 1828 rematch between Jackson and John Quincy Adams. Under Jackson's administration, enfranchis*m*nt was enlarged to all free (white) men regardless of means; until then, the right to vote had been limited to property owners.

During Jackson’s presidency, the United States evolved from a republic — in which only landowners could vote — to a mass democracy, in which free men of all socioeconomic classes were enfranchised.

Another Jackson achievement while in office was his dismantling of the Bank of the United States, again on the grounds that it was anti-democratic. One of Jackson's reasons for opposing this national Bank of the U.S. was that he believed it put too much power in the hands of a person who was not elected to office, effectively placing the head of the bank even above the president with regard to the economy. This was later proved true by Nicholas Biddle, the President of the Bank of the United States, when he drove the economy into recession to punish Jackson for his opposition to his institution.

Nicholas Biddle
At the request of Henry Clay and other Whigs, Biddle asked President Jackson to renew the bank's federal government charter in 1832. Jackson, a Democrat who held a deep hostility to many banks, declined to renew the charter, beginning a political debate known as the Bank War. When Jackson transferred the federal government's deposits to several state banks, Biddle raised interest rates, causing a mild economic recession. The Federal charter expired in 1836, but the bank was re-chartered by Pennsylvania. Biddle continued to serve as president of the bank until 1839.

Question:
If the U.S.was really meant to be a Republic . . .

If? Not only was the United States of America always meant to be a republic, but even after democratic reforms, the U.S.A. remains a republic to this day, not a democracy.

Five American Presidents have been elected after losing the popular vote:

• John Quincy Adams (1824)
• Rutherford B Hayes (1876)
• Benjamin Harrision (1888)
• George W. Bush (2000)
• Donald Trump (2016)

These outcomes were only possible because Presidents of the United States are not elected by a popular vote but rather by the College of Electors from each State, who are not necessarily directly or proportionately tied to the winner of the state’s popular vote.

While it is true that before Jackson, the “founding fathers” did not agree on much, one thing they did agree upon was that they were not creating a democracy, which they distrusted. The founding fathers were always clear that they desired a republic.

Originally the only branch of federal government left to be elected by the people was the Congress, i.e., the House of Representatives. The President was initially elected by the Senate, then to this day by the Electoral College, and not the popular vote. The Senators were originally elected by the State legislatures. The Supreme Court, the third branch of government, was appointed as it is today by the President and confirmed by the Senate. The people originally could only directly elect half of one of the three branches of government — in effect, “one-sixth” (1/2 of 1/3) of the nation’s leaders, the House of Representatives.

Democracies were seen as dangerous. In ancient Greece, a democracy had killed its greatest thinker, Socrates. A democracy tried to subjugate the Greek City States. A democracy was unable to protect its minorities from the tyranny of the majority. It was also perceived that ordinary people were vulnerable to being hoodwinked by political charlatans. The founding fathers instead favored a constitutional republic, which Plato had advanced to address the shortcomings of democracy. In this model of government, by keeping “men of substance” (i.e., those acclaimed as respectable, responsible, and reliable) as a steady hand on the wheel of state, with a constitution was the highest law, the rights of individuals are ensured to protect the minority from the tyranny of the majority.

Founding Fathers quotes on Democracy: .

Thomas Jefferson declared:
A democracy is nothing more than mob rule, where fifty-one percent of the people may take away the rights of the other forty-nine.

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Benjamin Franklin
Democracy is two wolves and a lamb voting on what to have for lunch.

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John Adams,
Remember, democracy never lasts long. It soon wastes, exhausts, and murders itself.

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James Madison, Federalist Paper No. 10
that democracies “have ever been spectacles of turbulence and contention; have ever been incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths.”

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Madison entire Essay ( Fed #10 ) is about why Republics are superior to Democracies and is worth a read.

Democracy vs Republic
The Founders never used the words "republic" and "democracy" interchangeably. They had studied various forms and systems of government from throughout history in order to establish a system of government that would best deter a tyrant (in their case King George III), or a group of tyrants, from denying God-given rights to Americans. Interestingly, the Declaration of Independence, the Articles of Confederation, and the Constitution do not use the term democracy to describe our form of government. Furthermore, "Neither the Articles of Confederation nor the Constitution set up direct democracies."9 The authors of these founding documents disagreed on many points, but on one point they ALL agreed wholeheartedly: The United States is not a democracy, never was, and never was intended to be. It is a Republic.

(Note: I have chosen in this answer to avoid the controversy surrounding the "Founding Fathers of the USA", who were certainly not of one mind, as choster has already explained in his fine answer, and stick to the US Constitution itself, as it was ratified and amended.)

Excellent question IMO, and there does indeed appear to have been something of a shift at some point:

Indeed, there is no mention of the term "Democracy" in the US Constitution or Declaration of Independence, or even an explicit provision for what we call "the popular vote" for President or for Senators. In the original body of the US Constitution, the election of the US President is given to the Electoral College, an institution that persists down to the present day, and the seating of Senators is delegated to the State legislatures:

President:

Article II, Section 1:

The executive Power shall be vested in a President of the United States of America. He shall hold his Office during the Term of four Years, and, together with the Vice-President chosen for the same Term, be elected, as follows: Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.

(Somewhat modified by Amendment XII, but the Electoral College remains)

Senate:

Article I, Section 3:

The Senate of the United States shall be composed of two Senators from each State, chosen by the Legislature thereof, for six Years; and each Senator shall have one Vote.

These provisions reflect the view expressed by Hamilton in Federalist 68:

It was equally desirable, that the immediate election should be made by men most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations. It was also peculiarly desirable to afford as little opportunity as possible to tumult and disorder. This evil was not least to be dreaded in the election of a magistrate, who was to have so important an agency in the administration of the government as the President of the United States. But the precautions which have been so happily concerted in the system under consideration, promise an effectual security against this mischief.

Contrast this view with Classical Democracy as exemplified by Ancient Athens:

Athenian democracy:

A political system in which the people do not elect representatives to vote on their behalf but vote on legislation and executive bills in their own right.

"Despised" is perhaps too strong a term, but it's clear that Hamilton and his constitution did not embrace classical democracy with open arms, preferring rather a "representative republic", closely resembling the bicameral structure of the British Constitutional Monarchy with "Upper and Lower Houses" - Lords and Commons - and the ancient Roman Republic in some respects.

It is difficult to pinpoint an exact moment or event that triggered the shift towards "Democracy", but a good starting point would be the 17th Amendment to the US Constitution, ratified in 1913, although not agreed upon by all the States.

Amendment XVII:

The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each State shall have the qualifications requisite for electors of the most numerous branch of the State legislatures.

States that rejected Amendment XVII

The following states did not ratify the Seventeenth Amendment: Utah (explicitly rejected)[38]Florida Georgia Kentucky Mississippi Rhode Island South Carolina Virginia.

Electing Senators directly rather than through State Legislatures is an obvious and dramatic shift in emphasis from the power of the States acting as proxies, to the direct representation of individuals in government - something akin to "Democracy".

However, the debate regarding the method for seating Senators had been raging for quite some time previous:

Roots of Amendment XVII

However, over time various issues with these provisions, such as the risk of corruption and the potential for electoral deadlocks or a lack of representation should a seat become vacant, led to a campaign for reform. Reformers tabled constitutional amendments in 1828, 1829, and 1855, with the issues finally reaching a head during the 1890s and 1900s. Progressives, such as William Jennings Bryan, called for reform to the way senators were chosen...

If we choose, we might go back further in time to Amendment XIV, ratified in 1868:

Amendment XIV:

Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

One the most important goals/impact of this amendment was to ensure enfranchis*m*nt for all former slaves in the post Civil War era. One could argue that this amendment caused the American political consciousness to focus more on the power of individuals in government and elections as opposed to government 'by proxy'.

We might also cite Lincoln's Emancipation Proclamation of 1863, which freed the slaves in the rebelling states of the Confederacy, and the victory of the North in the American Civil War, as factors which consolidated Federal power and diminished "States Rights" - the sovereignty of the individual states as opposed to the Federal government. Again, this made the individual more of a focal unit of power in the American Republic, than the States: a shift towards "Democracy."

This issue of States' Rights vs Federal power again goes back to pre-Consitutional days, was central to the American Civil War (Slavery and the vehement debates surrounding it that eventually resulted in war, were the result of the power granted to individual states to determine their own laws regarding slaves.) and has been an important theme throughout American history down until the present, with the general direction always being away from emphasis on States' Rights. This is manifest most dramatically (aside from the victory of the North in Civil War) in the many US Supreme Court decisions invoking the Commerce Clause and the General Welfare Clause to trump "States' Rights", and again effectively making the individual the most important unit of power in the electoral process, rather than the States (This is indeed a very lengthy discussion which I will not get into here.)

Along the same lines as Amendment XIV, we can cite Amendment XIX, ratified in 1920, which gave women the right to vote. Again, the power of the individual in the electoral process becomes a focal point.

Amendment XIX:

The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of sex.

There have also been numerous movements and attempts to eliminate or circumvent the Electoral College and move to direct Election of the President. This debate continues down to the present and was highlighted in the contested Election of 2000, when election results in Florida were disputed and that election's outcome came down to the question of which candidate could claim the electoral votes of Florida, although there was no question that Al Gore had won the majority of the national "popular vote". See Ideological Endowment: The Staying Power of the Electoral College and the Weaknesses of the National Popular Vote Interstate Compact - one example from a law journal culled from hundreds of references on this issue.

Again, it is difficult to pinpoint the exact time, events and reasons for these apparent shifts, although none of the issues are really new - they date back to Constitutional times, as we see from Federalist 68, the sources that choster has cited in his erudite answer, and many other sources.

My personal impression regarding "the Big Picture", taking into account all the above mentioned constitutional/legal shifts, is that as (virtually) universal literacy, education and suffrage became the norm in the USA, communication improved and more and more emphasis was placed on individual rights and liberties in the post Civil War era etc., as explained, Americans at large felt that they were all entitled to a direct say in the electoral process.

The result of this gradual change in American political focus and consciousness? Something closer to Athenian style "democracy" has become the projected image of the American Republic, and the term "Democracy" in modern American English is virtually synonymous with the representative republican form of government of the United States.

Still, that form of that government, in spite of the changes we have mentioned, remains quite different from that of Classical Democracy, as exemplified by Ancient Athens.

The question is poorly stated. The Founding Fathers were not all of one mind on many subjects— the Federalists saw danger in direct democracy, whereas the Anti-Federalists did not. Additionally, popular usage of terms like "democracy" or "republic" is quite different from a political scientist's use of such terms— indeed, quite a lot of things "don't mean what you think they mean," from liberalism to imperialism. Moreover, very few shifts in history can be traced to a particular "moment"; the world-historical figure as a character for study has been distinctly out of fashion for some decades. So, I will limit the discussion to the Federalist position on direct democracy versus representative democracy, i.e. a republic.

The entirety of The Federalist #10, by James Madison, is devoted to the question of direct democracy (which he calls "pure democracy") as opposed to representative democracy (which he calls a "republic"). It is arguably the most famous of The Federalist Papers. Madison argues that society is undone by faction, which he defines as

a number of citizens, whether amounting to a majority or a minority of the whole, who are united and actuated by some common impulse of passion, or of interest, adversed to the rights of other citizens, or to the permanent and aggregate interests of the community.…

A faction can be a minority, but also a majority, and the majority can work against the good of the whole by abusing its power. The majority simply cannot be trusted not to oppress the minority, given the opportunity.

If the impulse and the opportunity be suffered to coincide, we well know that neither moral nor religious motives can be relied on as an adequate control.…

From this view of the subject it may be concluded that a pure democracy, by which I mean a society consisting of a small number of citizens, who assemble and administer the government in person, can admit of no cure for the mischiefs of faction. A common passion or interest will, in almost every case, be felt by a majority of the whole; a communication and concert result from the form of government itself; and there is nothing to check the inducements to sacrifice the weaker party or an obnoxious individual. Hence it is that such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths. Theoretic politicians, who have patronized this species of government, have erroneously supposed that by reducing mankind to a perfect equality in their political rights, they would, at the same time, be perfectly equalized and assimilated in their possessions, their opinions, and their passions.

As an alternative, he offers

A republic, by which I mean a government in which the scheme of representation takes place, opens a different prospect, and promises the cure for which we are seeking.

The two great points of difference between a democracy and a republic are: first, the delegation of the government, in the latter, to a small number of citizens elected by the rest; secondly, the greater number of citizens, and greater sphere of country, over which the latter may be extended.…

The first allows that a body of citizens expressly chosen to refine the laws would be more focused and responsible in pursuing the true common interest over temporary or personal considerations. The second allows that direct democracy becomes unwieldy for large countries, but a republic would be efficient enough to govern while still encompassing a large number of interests, reducing the power of any individual faction.

Just read Federalist #10. The whole thing.