Past the typical definitions, and the biased definitions, let us look at the original Constitution (with the first 10 Amendments) and think.
A democracy is a system of government where the people choose who governs them. In a democracy, the majority rules, and what the majority wants is what the government must do.
A republic is a system of government where the people choose who upholds the laws that they have decided to live under. In a republic, the representative of the people (whether he be a president, senator or governor) is elected to follow and keep the laws of the state/commonwealth/country.
In the original Constitution, State/Commonwealth Senators to the government of the United States were appointed by the legislatures of the State/Commonwealth. The problems of majority/minority were non-existent because they didn't matter.
One example is in 1868, when Andrew Johnson was a hair away from being impeached. The House of Representatives impeached him by a landslide, but the Senate was trying to balance on a teeter-totter of the law vs. the people. In the end, only one vote decided it. The senators who dared voted against impeaching the president. Johnson's conduct may have been scandalous and he may have been a bad president, but there was no legal backing for impeaching him.
In my Internet search to verify the names of the dissenters, I came across this comment:
Well people, I hate to break it to you...but a democracy stinks. Majority rule is only one step away from the tyranny of the majority. Then comes the tyranny of the few...and that is only another name for socialism.
But the duty of the Senators, with whom we indeed differ, will be only the plainer. They must follow their sincere convictions, conscious that in so doing they maintain the only permanent principle of a free government; and their task will be the more difficult because they will maintain it against the cry of the party which is its natural protector.
It would probably be folly to try to assign a date to the day we became a democracy, but I think that the day the 17th amendment was ratified is close enough!