This is one of the reasons I find Rush Limbaugh so hard to listen to. He uses the term Liberal when he really means leftist. It’s really hard to take him seriously when he doesn’t use the correct terms.
In general, a leftist is one who prefers to see individual rights subordinate to government rights. A right winger is usually someone who believes in the rights of the individual over the rights of the government.
Orthogonally, a conservative is someone who wants to continue the status quo. Conversely a Liberal is someone who prefers to change things.
There are no correct answers here.
And my favorite example: the founding fathers of the United States were Right Wing Liberals.
In general, Marxists tend toward extreme left wing ideals. On the other end of the spectrum, anarchists prefer right wing views.
The Centrists (those who don’t have a strong left or right wing view) can be liberal or conservative. The “neo” tag simply means that they have chosen different issues than their past party has aligned with.
So the term neo-conservative has become an epithet much like neo-liberal. The parties have chosen a Left/Liberal, Right/Conservative line. The issues they chose to back supposedly fall somewhere on that diagonal axis. They prefer everyone to be in alignment even though the issues themselves often are not good liberal or conservative ideas.
An example of this is the peculiar tolerance for Islam expressed by some liberals. The political aspects of Islam are the very antithesis of liberal thought. Accepting people is one thing; swallowing their philosophy and religion as if there were nothing wrong with it is another. Everything the Liberal left has excoriated Christianity for, the misogyny, anti-gay, heteronormative preferences, and anti-abortion views, can be found in even more severe form in Islam, yet for some bizarre reason their political leaders accept it with open arms.
So it is not surprising that neo-liberals and neo-conservatives would question the alignment.
As for leftists who think socialism can work, all I’m going to say is that perhaps it does for a while. Unfortunately, much like any benevolent dictatorship, it may last for a generation or so, but inevitably it will degenerate in to a totalitarian regime. Centralizing so much power in government is a recipe for disaster. It is exactly why the US was created the way it was.
I sympathize with those on the left who want to help eliminate disparities and help the poor. There are many ways to do this. We don’t disagree on the outcome. We disagree on the methods. The goal is to find solutions we can all live with.