This is one of those things that’s technically correct but in colloquial conversation causes much confusion

No. The confusion is already there. The American population has been thoroughly confused. Using correct terminology is ILLUMINATING, and yes, it illuminates the confusion, and thus presents a chance to end the confusion. The American politicians LOVE the idea of two factions of liberalism controlling the entire discourse because then nothing makes any sense except the minor variations on liberalism that they manage.

They think liberal means left-wing. Socialist. Communism

They don't. They're not idiots. They're playing the population for fools. They know that they're fans of Adam Smith and John Locke, and they know that both of them were Liberal. They hire staffers from Harvard. Most of them are lawyers who studied this stuff. They know what they're doing. They're pandering and confusing. Deliberately. Because it gives them the power to manipulate.

Are there any major players in US politics you would identify as conservative?

Maybe Thiel and Yarvin who might actually think returning to monarchy is a good idea. But no. 99.9% of American Conservatives are Liberals and 99.9% of American Liberals are Liberals. It's a Liberal country founded on Liberal ideals by Liberals. It is the quintessential Liberal experiment.

The democratic party is also primarily concerned with private property and markets, with some lip service paid to civil rights and labor.

Correct. Because they're Liberals.

These ideas are not contradictory if you accept my thesis that the underlying concept is the preservation of hierarchy.

Which no one who studies the field accepts.

The idea is that there are some people who are on top, who rule, who have different rules applied to them

That's identical in Liberalism. Note how the Liberal founders of the USA, steeped in Liberal thought, made sure that the Senate was a body to protect landlord (or as James Madison put it, the opulent minority), how they made sure that slaves were private property and that slave OWNERS got more votes than merely people rich people who didn't own slaves, how women did not get to vote, how the states selected their senators and not the people, how they created an electoral college, how they made sure that polities were large enough to require large amounts of money and freedom to get elected into office. Liberalism was ALWAYS about protections for the rich and not for the poor. Always.

A republican senator can take a month of paid sick leave, because he’s Important, but a regular laborer cannot do such a thing. Sometimes it’s nobility. Sometimes it’s just the capitalist ownership class. But there’s also the lower levels, that are similar in a way. That’s where you get whites dominant over everyone else. Men above women.

Note. That's Liberalism, my friend. Racism gets codified around the time maritime mercantilism is getting incubated. By the time Liberalism becomes a philosophy, it's fully invested in racial hierarchy. When the Liberal revolution happens in the US, the founding medical universities were all pumping out race "science" to justify slavery. That's ALL Liberalism. Kings didn't make people racist. Merchants did, because they needed to prove to courts of law that throwing slaves overboard wasn't murder, it was property destruction and thus covered under insurance claims. Because that's how Liberalism works.

White supremacy is not a Conservative concept. Patriarchy is not a Conservative concept. Were there Liberals who wanted to abolish these things? Yes. Is it because they were Liberals? No.

The confusion comes from the propaganda of Liberalism as a defender of "universal rights", when what it really did was invent the concept of abstract "rights" as way of fighting the "injustices" of monarchy, when those injustices were specifically "I can't buy land because I'm not nobility". The only reason they were expanded to include things peasants cared about was to get the peasant to fight on the side of Liberals. But the Liberal philosophy drove the absolute height of chattel slavery in the West, complete with it's utter brutality and it's Christian nationalist mythos.

I’ll submit this suggestion to my book club. (No, I’m serious. We just read Zinn’s people’s history. Don’t laugh.)

Zinn is great, but it should absolutely make everyone pause and go "wait a minute, we don't actually know what we're talking about" and then reading Locke and Burke become salient. And then when you read them, I swear you'll see things in Locke that you'll realize are totally batshit, like the idea that if a virgin forest is inhabited by indigenous humans who forage for food, they don't own it, but if I personally chop down a tree, sell it for money, then hire 2 people to chop down more trees, sell them, and then pay an army to chop down all the trees, now I and I alone own all of that land and it's within my right to kill anyone who would trespass that land AND ALSO the purpose of government is to provide a police force that will kill people for me because I couldn't POSSIBLY ever personally defend all of the land that I would be able to own....

And then you'll read Burke and realize, holy shit these Conservatives literally had a social welfare perspective that the nobility was required to provide for the common people, and then you'll really understand the People's History because after nearly every Liberal revolution in Europe there was like a full century of just fucking economic devastation where the merchants took all of the wealth and the laborers lived like cockroaches (and died like cockroaches) and the Liberals were like "work harder, and smarter, and maybe you'll earn a life one day" and that shit went on for so long it literally created Marxism and the Liberals were suddenly fighting against organized labor and they saw the revolution in Russia and were like "holy shit we're gonna die" so they reinvented social welfare to protect themselves.

I know it's confusing to Americans. But that confusion is not my fault, and we have to work through the confusion for people to finally understand what the fuck has been happening all these years. It's two factions of Liberalism, primarily separated by the particular psycho-social profiles they resonate with the most, and they collaborate to maintain the Liberal power structure against liberatory movements that would unseat them from power.