I really don't get that people don't understand this. I mean, the term republic just means that the people vote for the leader they want. It effectively means that you don't have a monarch. A democracy is a country where all the people vote for that person.
I wouldn't say those are the modern meanings of the terms. Various countries have people vote for who they want to despite having a monarch. And one can't say a democracy is a country where all the people vote for someone, because in no country does every single person have the ability to vote.
I would say I agree with this definition of democracy from the New Oxford American Dictionary was being a good one:
"a system of government by the whole population or all the eligible members of a state, typically through elected representatives"
Its definition of republic, incidentally, is:
"a state in which supreme power is held by the people and their elected representatives, and which has an elected or nominated president rather than a monarch."
One may notice there is considerable overlap between these.
When people try to insist on the whole "the US isn't a democracy, it's a republic!" they are insisting on using
old definitions for these terms. It is true that in the past, the term "democracy" had a more restrictive meaning, and a republic was seen as in opposition to democracy. Here's the definition of democracy
offered in Samuel Webster's 1828 dictionary, which I believe is the first major American dictionary:
DEMOCRACY, noun [Gr. People, and to possess, to govern.] Government by the people; a form of government, in which the supreme power is lodged in the hands of the people collectively, or in which the people exercise the powers of legislation. Such was the government of Athens.
Now compare the same dictionary's definition of republic
REPUB'LIC, noun [Latin respublica; res and publica; public affairs.]
1. A commonwealth; a state in which the exercise of the sovereign power is lodged in representatives elected by the people. In modern usage, it differs from a democracy or democratic state, in which the people exercise the powers of sovereignty in person. Yet the democracies of Greece are often called republics.
2. Common interest; the public. [Not in use.]
Republic of letters, the collective body of learned men.
This explicitly draws a distinction between a republic and a democracy. A republic has ruling representatives elected by the people who rule (indirect rule by the people); a democracy has the people rule themselves (direct rule by the people). The example it gives of a democracy is ancient Athens, in which everyone eligible (namely, anyone who was white, free, adult, and not foreign) could speak in the assembly and vote on legislation. Basically, the government was essentially run by referendum. That was what a democracy was viewed as, in comparison to a republic where people elected representatives who did all of that.
So this attempt to draw a distinction between democracy and republic isn't out of nowhere. The problem is that when I look at the calendar, I see 2025, not 1828. The fact people distinguished between those terms 200 years ago in this way does not mean that is the meanings of them now. Since then, the term "democracy" has expanded considerably in meaning to include systems with elected representatives, and thus now rather than democracy and republics being different things, a republic is a kind of democracy.
Indeed, even in Webster's time the term democracy was being used to refer to republics like the United States, even if it was not as common as it would be later. For example, consider
this remark by John Quincy Adams in his inaugural address from 1825, a few years before Webster's dictionary was published:
"If there have been those who doubted whether a confederated representative
democracy were a government competent to the wise and orderly management of the common concerns of a mighty nation, those doubts have been dispelled"
As another example of word shift, consider the following definition from Webster's 1828 dictionary:
AWFUL, adjective [awe and full.]
1. That strikes with awe; that fills with profound reverence; as the awful majesty of Jehovah.
2. That fills with terror and dread; as the awful approach of death.
3. Struck with awe; scrupulous.
A weak and awful reverence for antiquity.
Shakespeare uses it for worshipful, inspiring respect by authority or dignity.
Our common people use this word in the sense of frightful, ugly, detestable.
This indeed shows us a shift that was already in progress, with its note of how while Shakespeare used it for worshipful (fitting with definition #1), "our common people" use it in the sense of being frightful (#2). The word awful comes from "awe" and "full" to mean full of awe, from which we see definitions #1 and #3. It at some later point developed definition #2, meaning frightening. But none of these fit to its most common modern usage, which is to mean unpleasant or very bad, which probably derives from #2 but is still rather different.
Thus this argument I think is generally irrelevant. The one point in which the republic/democracy distinction might matter is if someone is arguing "we're a democracy, so the majority opinion should rule"
then the republic/democracy distinction might be relevant, but even there it would be more accurate to say the United States is a democracy, but not a
pure democracy, which means majority doesn't automatically win everything. However, I don't think that was at any point an issue of discussion here; whether the United States is technically a democracy or a constitutional republic appears to be irrelevant for the purpose of the question that was presented in the topic.