Hi Raffey. Honestly, I can't set you straight. You seem to know as much about it as I do.
To me, it's the idea of small government and more active private sector. Which I used to advocate myself. I remember often saying 'oh government is so inefficient, they should just get out of the way and let people do stuff'. Since then I have worked in business and I realize that business are, if anything, MORE inefficient, you just don't see it because businesses fail at such a high rate and only the functional ones survive.
I think this confusion about who 'owns' neo-liberalism may be because of this.
The US does not actually have a left-wing party. Just two right-wing ones. The Democrats are actually WAAAYYY to the right of almost anywhere else in the world. For example, we have actual communists in our Parliament. Bernie Sanders would be considered right wing here, and even in much of Europe. If someone came around talking about free healthcare people would be like 'and... what?'
The Democrats have also been weirdly free of ideas. They just take Republican ideas and try to make them marginally less evil. So while the genesis of neo-liberalism could be said to be Republican (and the idealogy most eloquently expressed) by Reagan, Democrats have also carried it out.
The Democrats never really came up with an alterate governing philosophy. Clinton's 'triangulation' was this. For example, Obamacare is actually Romneycare from Massachusetts. It's a Republican idea. They didn't change the paradigm, they just tried to work within a Republican frame.
Which I think is a problem, especially since the Republican do not give a fuck. They won't even vote for their own ideas, they increasingly just want power to loot and enforce white supremacy.
So, yes, in many ways Clinton and even Obama have ended up owning neoliberalism. I think this may make sense if you see that America does not have a right and left party but in fact a right and fascist party. Both parties are neo-liberal.