Capitalism is an economic system in which trade, industry and the means of production are controlled by private owners with the goal of making profits in a market economy.
This where it gets tricky though. There are only two incompatibilities between my preferred version of socialism and the definition of capitalism you posted:
1. Property is collectively owned by the People but its management contracted out into private hands courtesy of constitutional provisions - not directed by the state itself.
2. Firms must be controlled and democratically managed by their workers.
Socialism is not synonymous with state socialism or even Marxist thought. I am fine with rigourous competition, profit motive, people earning different levels of income, and a great many other things traditionally associated more with capitalism than socialism.
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