So. To your definition, can there be representatives in a democracy? How many cav there be? Can there be a president in a democracy? Can there be made laws without popular vote un a democracy? How many people within a space can be excluded from voting and it is still a democracy? By what rules, you have the right to vote?
When all these question are not answered just the same by most people. Then probably there are different types of democracy. Democracy is a cluster of states, that agree, that there is a group of people which can vote for the lawmaking organs.
So it would be silly, to think that a democracy is a very clear norm. Just look at how the USA, Germany, France and Switzerland implemented democracy.
Which one is the only correct democracy in your idea?
Not even in Monarchy you would be true. Great Britain, Thiland and Liechtenstein are all Monarchies. Are they the same? In no way. Liechtenstein has a similar democratic system as Switzerland has. Still they have a Monarch, that can decide on almost everything. The speciality is, that the people of liechtenstein could fire him from office, whev they do not want their monarch anymore.