I haven't redefined anything.
> A republic is just that, a form of government where the people are represented by elected officials/representatives.
In 1999, Australia had a national referendum on becoming a Republic [0]–to replace the Queen with a President. The referendum was defeated, despite polls showing a majority of Australian voters favoured becoming a Republic. Most observers attribute the referendum's failure to disagreements among Republic supporters over how to elect the President – many wanted a directly elected President, but the model proposed by the government had the President elected by the Parliament instead. Advocates of the indirectly elected President model were concerned that a directly elected President would become a political office, and transform Australia's political system from a Westminster-style parliamentary one into an American-style presidential system or a French-style semipresidential one. (The fact that Ireland has a nonpartisan directly elected President while retaining a Westminster-style parliamentary system did not seem to be considered.) Also, some were concerned about the fact that the government's proposal gave the Prime Minister the right to sack the President at any time for any reason, although if sacked the President would be replaced by the longest-serving state Governor (which is a non-partisan position) not by a candidate of the Prime Minister's choosing.
If "Republic" means "a form of government where the people are represented by elected officials/representatives", what on earth was Australia voting on – they already have that. The fact that Australia already has "a form of government where the people are represented by elected officials/representatives", and yet isn't a Republic, is pretty clear evidence to me that is not what the word "Republic" means.
> To use a real world example you don’t find many historians that call the ancient republic of Rome a democracy, or a representative democracy.
The Roman Republic was really closer to a plutocratic oligarchy than a representative democracy. The system was set up so the ruling class (the patricians, and even the wealthy plebeian families) had far greater representation than the poor free masses (to say nothing of the slaves). It was definitely not a system of "one person one vote" (or even "one man one vote"). What made it a Republic then? It became a Republic when they abolished the Roman monarchy. It effectively stopped being one when the monarchy was de facto reinstated with the Emperors, although in the earlier Roman Empire, the Emperors still retained the pretence that they were merely "first citizen of the Republic" rather than monarchs (the Principate); it was only in the later Empire that pretence was dropped (the Dominate).
Actually, the early US Republic wasn't a representative democracy either, it was a plutocratic oligarchy like the Roman Republic was – most people were denied the right to vote. Many states imposed voting restrictions based on race, gender, literacy, property ownership, etc; generally speaking, only middle-to-upper class white men were allowed to vote, and representation was very often denied to women, non-white men, and poor white men. It was really only in the 20th century as those voting restrictions were lifted that the US finally became a genuine representative democracy.
[0] https://en.wikipedia.org/wiki/1999_Australian_republic_refer...