Your definition of private property is the definition of personal property. Marxist understandings of private property = ownership for private interest of the products that are the means of producing profit and, on a practical level, the vast majority of the wealth of the nation. This reaction is as old as the Communist Manifesto - in fact, it's older, because the CM actually takes it on and responds to it.

As for the quotation, it in no way surprises me Churchill says this; he is the author of the "Iron Curtain" metaphor, and a major propagandist for western capitalism. He was the head of the Conservative Party in Britain - pro-empire, pro-capitalist, anti-labor, pro-free market ideology.

Socialism may come out of a certain ignorance about human nature

I'm sorry, I'm going to have to say something here. In my experience, people who say this haven't considered seriously the critique of human nature leveled by Marx and Marxian theorists, which is a critique against philosophical notions that simply reify the dominant behavior of people without looking at what promotes and causes those behaviors to be dominant. For a socialist who wasn't sidetracked by Fourier, St. Simon, or anarchism, it's not a question of eliminating evil in the hearts of men and remaking human nature absolutely, but of determining what causes the potential in all of us for massively anti-social behavior to become a dominant force in society. There will of course always be people who are morally bankrupt; that potential always exists, but even four hundred morally bankrupt people, the number of people who currently own as much wealth as half the planet - literally - if they are not in control of production and distribution are not going to be able to have such a significant impact on society as they do when they have power thanks to money.

That's the socialist bet.