So we would be forced to conclude based on this definition that the UK is both a monarchy and a republic, which seems odd.
So long as the nominal supreme authority lies with a monarch, even if that authority is as thin as the paper it is written on, the UK would remain technically not a
monarchyrepublic. Indeed, those who seek to do away with the monarchy refer to their political position as "republicanism" and refer to replacing the monarchy with a republic.
However, the UK is a republic in all but name. The same would apply to other constitutional monarchies, at least those where the nominal power of the purported monarch remains effectively ceremonial.
The United States, commonly regarded as a republic, is also certainly a democracy. In fact, there are more democratically elected offices at the local, state and federal government than, perhaps, in any nation in the world.
The Constitution also contains numerous actually anti-democratic features specifically to guard against the worst problems of a so-called "pure" democracy, which would amount to mob rule, a form of government which has, in fact, never actually existed. If only a "pure" democracy constituted a democratic form of government, there would be no such thing on the planet.
Among those anti-democratic features are the Article V process for amending the Constitution itself, which essentially puts it in a lockbox of sorts beyond meddling by public whim. Only a sustained political will over a lengthy period of time can change the Constitution itself.
Additionally, having unelected, lifetime appointed judges have supreme authority over interpreting that Constitution stops radical changes from overtaking interpretation. Despite this, even the most "activist" decisions of the Court generally follow rather than precede public opinion.
ETA: Correcting mistake.