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.
The Kurds are an exception, arguably should have their own state and would be the most sane partner in the middle east, but the US can't get its shit together to stand up to Turkey. Alas, we routinely screw over the Kurds as they get gassed by Saddam Hussein, ethnic cleansed by Turkey, abandoned to destruction by Russia when they were our best anti-ISIS ally.
Oil money and the wealth inequality that came with it certainly don't help things to engineer functioning democratic states, and then as stated elsewhere, neither does the CIA toppling democratic governments because multinational corps find them inconvenient.
America building "democracy" in Iraq was a telling process. All we cared about was oil and maintaining political control. We didn't care about making the lives of the everyday person better, which is the true fundamental path to a functioning democracy (it's why the USA's is gradually apart after all).