> A republic doesn't just mean no monarch.
Yes it does.
> A republic is different from a pure democracy because it explicitly protects the rights of minorities through a constitution.
Not true; a constitutional system does that, but that has nothing to do with being a republic. Many republics have no such protections. You're conflating a constitutional government with a republic. Republic means nothing more than no monarchy. There are lots of republics in the world, they don't all protect minorities through a constitution or even have a constitution because that's not what the word republic means.
A republic is a society in which the interests of the people are represented by elected or appointed officials who do not inherit their positions.
Madison really liked the term as a way of distinguishing Roman democracy from Athenian, but there's no hard definition, or in practice there hasn't been.
Ancient Rome, the UK, Venice, North Korea, the US are all "republics".