You can find socialism and capitalism defined in very similar ways to how I defined them in this thread in dictionaries, encyclopedias, and university textbooks in economics.
For instance, the Encyclopedia Britannica defines socialism as a "social and economic doctrine that calls for public rather than private ownership or control of property and natural resources." While it defines capitalism as an "economic system, dominant in the Western world since the breakup of feudalism, in which most of the means of production are privately owned...." Those are very similar to how I defined the terms earlier in this thread.
As an economic system, the crucial difference between socialism and capitalism is who owns what?
People are free to define terms anyway they want. There is no god nor law of man or nature that states the "correct" or "proper" use of a word. To argue that you have the one and only true definition of a word is poppycock. In this thread I am going with the most common meanings of the terms socialism and capitalism.