After reading Apollo, I need to correct some definitions here.
Leftist- you like equality. You might want to government to redistribute, or you might be an anarchist who wants a decentralized but egalitarian society, but as long as you want equality, you're of the left.
Rightist- Authoritarian is the usual description, but you're starting with the belief that a certain level of inequality is good, justified, or necessary. How much inequality you want is yours to decide.
Centrist - what you get called when the media wants to praise you for appearing to be in between two arbitrary points. In the center? The center of what? Where are the two extremes located? Mussolini's politics were somewhere between Hitler and Stalin. I guess that makes him a centrist! You can be in between something literally anywhere on the left-right spectrum. It means absolutely nothing. It's a marketing term. Used primarily to create the illusion of disagreement that requires a centrist to go and 'solve'.
Liberal - you think people should be free(er). You might advocate this by having the government get out of your business, or you might call for government intervention to prevent others from violating your civil and economic freedoms. You might focus on negative liberty (free from coercion) or positive liberty (free to fufill one's potential, unburdened by the cost of education, for example), but you just have to focus on freedom. Of course most people who actually take freedom seriously in this era consider both the oppressive power of the state and the oppressive power of concentrated wealth, but there's plenty of room for disagreement. Liberalism is a big tent to those who take it seriously (ie, not Democrats).
Conservative - you believe in traditional values. So, in other words, you happen to be in favor of whatever you think are traditional values. If these traditional values include say, liberty and freedom from having to subject yourself to wage labor (like say, THE PLATFORM OF THE REPUBLICAN PARTY WHEN ABRAHAM LINCOLN WAS PRESIDENT), you can be a conservative and be the most ardent left-liberal at the same time.
Most of what you say is correct. However, note that most leftism uses the power of the state to force a measure of equality. That's what I mean when I say they're directly proportional.
All other things being equal, a capitalist society will be freer than a socialist society.
EDIT: Oh, and a rightist who wants equality but hates government intervention more is still a rightist. A democratic state can also be right-wing.
No, someone who wants equality but hates government intervention is a left libertarian. Right and left have nothing to do with freedom directly. You might argue that equality is a necessary precondition for freedom, or that equality in the real world requires the government to violate our liberty, but again, that's up for discussion. The definitions are not. Left = equality, right = not equality.
Here are some more definitions for you:
Socialism- An economic system where the workers run the workplaces (especially the factories) and make all of the the economic decisions, including for investment. Notice the lack of mention of the government. Government ownership of the economy is not socialism, regardless of whether or not the government claims to own it in the people's name. State socialism is a contradiction in terms, invented by Lenin and Trotsky to legitimize their dismantling of the democratic worker organizations that sprung up in the wake of the Russian Revolution, and the concentration of all power in their own hands. The west was happy to call this socialism because it equates socialism with one of the horrors of the 20th century and not its real meaning, the democratic ownership of industry. Socialism is workers control.
Capitalism - An economic system which separates society into workers and capitalists. The capitalists are endowed with control of the physical production capital of the society and make the crucial decisions of when and what to produce by way of private investment. The funds to invest are provided by a banking sector with the power to create purchasing power
ex nihilo to award to entrepreneurs, allowing the economy to (potentially, if this power to create credit is not misused to inflate the price of existing assets instead of creating new ones) grow faster than would otherwise be possible if investment were funded only by existing income. Workers are paid wages for operating the capital and (at least nominally) influence production by deciding what produced goods to purchase with their wages.
Free market - a term with any number of definitions. By the original definition given by Smith, Ricardo, and John Stuart Mill, a free market is a market free of any economic rent, be it caused by government interference or by private exploitation of monopolies. They key here is that in an economy free of rent, prices fall to their floor - the cost of production, because there is no additional overhead businesses must pay to produce a good beyond the direct price of the labor and capital used in its production. The irony is that classical liberals actually believed that the government had to intervene in some way to remove rent from the equation - the only question was how to do it while minimizing the side effects to the rest of the economy.
The popular modern definition of a free market is an economy with literally no government interference. One type of overhead is eliminated, while the other is allowed to grow freely. Such a free market has never existed, ever, certainly not in western societies, which all developed with heavy state intervention (who do you think paid for the development of the first computers? Or the internet? Certainly not the private sector). However, such a definition is used as a positive strawman to justify the actual economic system in which we live, in which the development of new technologies is paid for with public funds until said technology can be sold at a profit, at which point the patents are carted off to private sector monopolies who can sell at high prices to consumers who already paid for the development costs with their tax dollars, thus ripping them off twice.
Also notice that the original definition of free market does not require the concentration of the means of production in relatively few hands, as in capitalism. In fact, a believer in the free market might consider such an arrangement hostile to the free market.