There is a difference between words changing over time (example: awful historically meaning full of awe and the modern usage meaning very bad) and changing a form of Government of the US that is codified in the US Constitution.
>If not, then what are the differences between republic described by Madison and representative democracy?
To try to answer your question, if we used “Representative Democracy” then the US and UK would both have the same form of government, a Representative Democracy, which in my opinion highlights why proper use of the terms is necessary.
The US is a Constitution Republic and the UK is a Constitution Monarch, at least according to the legal documents that establish their respective Governments.
Whether a Constitutional Republic or Constitutional Monarch, neither is a Democracy. Sure both having elections and voting of at least some kind, but the UK is not a Republic despite having a House of Commons with elected Members of Parliament.
Let’s turn this question around…in your opinion what is the difference between a Republic and a Democracy? If any Republic that holds elections of representatives do you simply classify both the UK and US Representative Democracies?