Liberalism literally means supporting liberty, meaning the basis of thought is reasoning around the concept of liberty and equality(though classical focuses more on the liberty aspect while the social variant focuses on equality). This is in contrast to conservatism, which functions on the basis of experience with tradition and authority.

Conservative and Liberal are political philosophies. Left-Right wing is political classification(liberals fit under the left, conservatives fit under the right). Then with economic principles, there is a slightly different spectrum due to how those are defined in relation to political classification and philosophy.

You can break it down to social and economic, but in the US, liberal(Democrat) economics generally means regulated capitalism while conservative(GOP) economics refers to free market capitalism. Specifically, the Democrats/liberals are social liberals regarding the economy while the GOP/conservatives are economic liberals on the economy. It's confusing, but a major reason has to do with the fact that the US is technically a spectrum of liberal on the mainstream level(because whether we talk about liberals or conservatives, free market capitalism is supported but with opinions split on how regulated it should be and how it should contribute).

On a philosophical level, liberal is the opposite of conservative. A liberal supports liberty and relies on reasoning to support their conclusions. A conservative supports tradition and relies on experience to their conclusion. They're both direct opposites that hold different approaches on what to base decisions on.

Neo-liberalism is nothing more than economic liberalism taken a step further with an intended goal of privatizing everything(rather than mere deregulation and barrier removal). In terms of the US, it's predominantly something followed by the GOP rather than the Democrats.

A reactionary is basically a "conservative progressive" in the sense they want to move in the opposite direction. Reactionaries are generally considered to be "extremely conservative" as they do not want to merely resist frequent reform/progressivism, but actually adopt a status quo ante because of the belief that the previous status quo was functional and effective.

The reason this isn't as well understood in the US and the mixing of "leftist" with "liberal" and vice-versa is because the US political center IS liberal as we have no historic conservative base. The only true conservative stance would basically be believing the US should go back to being a subject of the United Kingdom and I don't think I've heard of such a belief being popular after the Revolution.

Our conservatives are just less liberal than our liberals. It's literally a case of "left-liberal" vs "right-liberal". In the end, we're all liberal because the US is still relatively new as a country and because the US formed as a liberal country on the basis of liberty, freedom, and equality from the start. We were never a monarchy, so we never developed the kind of paternalism found in other parts of the world. It's also why we have such a hard time adopting "foreign" political ideologies that don't reside within our boundary of liberalism.