First, I am fairly sure the government that uses representative democracy is called a Republic. A classic Republic was Rome but it did not have a Constitution to limit the representatives. The founding fathers decided to add a Constitution to severely limit what the federal government could do. Of course, it only works if the representatives follow it. THus, the US’s federal government is a Constitutional Republic which is executed in representative democracy, at least in part.
Concerning misuse of words, anyone can start using a word to mean the opposite of what it is defined to mean which is your point about the misuse of the word republic today.
The word liberal was changed about a hundred years ago when leftists started taking over our federal government. I call myself a classic liberal but if I use that term today, most people will conclude I am a democrat, progressive or leftist when in fact, i am the exact opposite.
Here is a Friedman video on classic liberalism.
https://m.youtube.com/watch?v=QmeeMYrnweg
JoMa
The term democratic-republic covers what you mention.
Membership in the Roman Senate was based on being a member of the elite (patrician) class and also how much money you had and land you owned.
Here’s a list of countries with a written constitution:
https://en.m.wikipedia.org/wiki/List_of_national_constitutions#Codified_constitutions_(most_recently_adopted,_remaining_in_use_today)