Saul Kripke needed one hundred pages to show some difficulties with using the word “Aristotle” in our language.
Politicians had similar problems with common nouns. When Jefferson mentioned “All men are created equal,” he didn’t mention that “men” meant white men with a fortune.
“Democracy” eclipses other words by the number of alternative meanings. Churchill and Lenin imagined totally different things when they used it. And so did thousands of other authors. Good luck finding two authors who meant the same by the word.
But economists have to put democracy in equations. Words fit equations badly, and you need a number. A number for “democracy” usually comes from Polity IV, World Bank, Freedom House, and other organizations. The numbers have been even brought together in a single dataset.
And yes, these numbers still mean different things because organizations compose them from different facts. Here’re Polity IV’s democracy components:
Competitiveness of Executive Recruitment
Openness of Executive Recruitment
Constraint on Chief Executive
Competitiveness of Political Participation
Still, an indicator consists of facts: whether elections are competitive, presidents have to ask permissions, or country has elections at all. This indicator can jump into equations and help understand relations with other facts, like wages. In the end, you may be interested in the outcomes of democracy, like better living. Indicators happen to be very specific about this: if you had this-and-that variable equal one, then your income would be 20% higher.
A huge improvement over Churchill, who said basically the same thing, is that numbers can prove it. As far as democracy means a fixed list of facts, these facts can be systematically studied. On the other hand, storytelling about democracy proves nothing, it only convinces. People got convinced by various lies for centuries. That’s what history is about. And that’s also why you need a mechanism called science to check what happened to be true after all.