@freeschool @freemo
Sure, I agreed with you. A few remarks.
You say we don't have a free market now, I guess you refer to the United States - don't know about all places, but I feel a purely free market capitalism is practically unachievable unless you go into an anarchist society. As long as the government exists the government will be doing something: they must get some money somehow. It may not impose tariffs or trade restrictions but you can argue that any kind of taxation breaks the free trade in one way or another.
I feel the Oxford dictionary definition covers capitalism quite well and in a short sentence "an economic and political system in which a country's trade and industry are controlled by private owners for profit."
I would not use plutocracy as a synonym of capitalism at all. That would imply that you cannot have a democratic state with a capitalist economy. Sure, democracy is strongly influenced by the rich in capitalism, but they don't necessarily have government powers and their interests can be disregarded by politicians.
I would not include state capitalism as a subtype of capitalism. It is in fact very different from capitalism, since individual people do not own the means of production and you don't organise production to generate profit but rather to supply the population with certain things. The decision process of which things to produce is very different from capitalism, you don't have a bunch of people deciding what the economy should produce according to how much profit they can make, but rather politicians and economists taking decisions according to what they feel is better for the nation.