I disagree that Muslims want democracy (I'm Muslim). Present day democracy generally contradicts and is against Islam, so I'm not sure where you're getting your information from.
Shadi Hamid has been instrumental in helping me understand the distinction between the two things. Here’s a good example: https://mobile.twitter.com/shadihamid/status/144363599537580.... More generally: https://slate.com/news-and-politics/2016/08/shadi-hamid-on-i...
What (most) Muslims want is something like what America was like in the early days. Puritan society was extremely democratic, insofar as the people did make the rules for society. But it was not liberal democracy—the puritans created public schools for the purpose of socializing children into religion. They also made it a criminal offense to celebrate Christmas: https://www.history.com/news/when-massachusetts-banned-chris...
But which rules? While Islam does have certain leeway for certain things to be left to society to decide, not everything is. You will not find any Muslim who will make it an offense to celebrate Eid for example.