The text of Federalist 10 shows that this is not a useful distinction in our contemporary context. Madison defines the "republican principle" as majority rule, and further defines "pure democracy" as direct government by an assembly of the citizen body, i.e. Ancient Athenian Democracy, something which does not exist at any scale in the contemporary world. A republic, as Madison defines it, is one in which the citizen body elects representatives, who then carry on the work of government. In modern parlance we call this a representative democracy. Further, Madison emphasizes the vital importance of majority rule in a republic as a guard against faction, so much so that he call majority rule the "republican principle" as noted above.

The bottomline is that the essential features of what Madison calls a "republic" is what the contemporary world calls a "democracy" or more specifically a "representative democracy". Insisting on calling American government a republic and not a democracy, when Madison's definition matches what we think of as a representative democracy, just obfuscates what's being discussed.