The US is a republic, the UK is a monarchy, but both countries are representative democracies, they describe different aspects of the political workings of a country and are not mutually exclusive. I think you are confused by the definition of democracy, giving it too narrow a meaning.
http://www.merriam-webster.com/dictionary/democracy
I know some people on the web argue what you are arguing but they are wrong.
http://www.merriam-webster.com/dictionary/democracy
I know some people on the web argue what you are arguing but they are wrong.