* liberal here meaning political liberal (JS Mill liberalism), and not what is understood by "liberal" in US politics.
There is an overlap between democratic republics (or, to use the less common phrasing, “republican democracies”) and limited representative democracies, but they aren't the same.
The US Founding Fathers copied England, and Rome's model but replaced the sovereign with a President.
Representative democracies are necessarily republics - although some, like my country, skip the upper house. Rome introduced consuls / republican model because the population was too large, unlike the Greek polis, to be politically manageable. Rome of course was not a democracy.
*or monarchic. I am away from my books so cannot confirm exactly the term. Sorry.
Republican Rome was distinguished from both preceding and succeeding Roman governments by not having a sovereign head.
It's a constitutional monarchy based on a parliamentary system.