FWIW:
I've always considered a Republic as a 'subclass' of a Democracy.

As the USA is a Republic, it is also a Democracy, just a special case of a Democracy.

I do think that a Democracy is the rule of the people... how that rule in 'implemented' is a 'detail' In some it is by direct voting ( State petition, members of congress etc. ), others via representatives ( Votes in congress ) another way could be elected monarchs, e.g. Star Wars.

Anthony