Essential and the most important part of modern democracy is freedom: Universal, individual human rights. It is the foundation of the US: 'All men are created equal and are endowed with inalienable rights ...', the land of liberty, home of the free, liberty and justice for all. The Constitution's limited remit to government, and the specific limits, are about freedom and self-determination.
And that's certainly liberal - that is the same foundation of classical liberalism and modern left-wing liberalism.
It's absurd to suggest otherwise; it's pointless arguments about verbiage. Nobody wants mob rule, nobody thinks of democracy without human rights and indvidual freedom, either scholars or the general public.
> It’s about people being able to vote to live in the kind of society they want.
Why should they get to decide en masse what people do individually? That's just dictatorhip of the majority, of the mob. Freedom is what the US and democracy are about.
> Democracy is not about free trade, or protecting immigrants, or the like.
It is; it's about freedom.