There has not been a true democracy since ancient greece for a state, though numerous tribes in europe have tried it in regions during roman times. Democracy is a failed idea, it is always the majority voting to oppress the minority. A republic is representative democracy, it takes the better ideas of democracy but adds a direct layer between the people and the laws, but also holds govt more accountable to the people.
It is not the most ideal system, but it is the best of what we have, and has done better at protecting minorities than other systems have excluding some dictatorships, however minority protection is not uniform among dictatorships either.
What you call "True democracy" is actually known as "Direct Democracy" and is impossible to implement in all but the smallest of communities and yes that included ancient Greece too.
What we have is "Representative Democracy".
The UK, Canada, USA, Australia etc are all representative democracies - differing only in the details on how representatives function.
In a republic the head of state is elected
In a constitutional republic like the USA, the head of state (president) is elected by the people rather than by a smaller group of people holding political power.
So the
USA is a democracy (if you don't believe this, then ask the US government itself)
The people don't vote on individual issues but instead elect
representatives (the reason the "House" in Congress got its name Btw) - so the USA is a
Representative Democracy (again confirmed by the US Government)
The head of state is elected and conforms to a set of rules/laws (the Constitution) - so the form of representative democracy in the USA is one of a Constitutional Republic
Canada and the
UK are slightly different:
Both are democracies - like the USA
Both elect representatives to sit in their respective legislatures and so are Representative Democracies - just like the USA
However both have a head of state who's claim is by birthright not election...they are therefore Constitutional Monarchies
the people of the USA and Canada/UK are no better protected than each other.
What protects the people are laws and the judiciary.
Those laws do NOT require a formal constitution and neither do they require the head of state to be elected.