This is beyond old.
What are the main types of democracy? Meaning of democracy and its types. Five types & forms of democracy with examples. Types of democracy.
bscholarly.com
The US is #2 and #4.
Democracy denotes the source of power (the people), along with other characteristics (rule of law, e.g.). Democracy is
how we choose our government.
A federal republic is a
form of government. #2 in the list, as opposed to a parliamentary system.
Since you're lost on the difference between methods of forming a government and the structures a government can take, I won't waste my time on the rest.
This tired old trope is a sign of poor education. Best to keep it under your hat.