A big part of the confusion comes from the fact that different people will use these terms differently.

In a capitalist framework, there's private property and public property. Either an individual (or or specific group) own something, anything, or it's owned by the government.

In a socialist framework, private property is distinguished from personal property. Personal property is your stuff that you use for yourself. Your coat, your car, your TV, etc. Private property is the means of production, or capital—things that increase a worker's ability to do useful work. Think factories or companies, where ownership in and of itself, regardless of labor, would make the owner money. Socialists think that kind of private property shouldn't exist, because it means wealthy people can just own stuff for a living, profiting off of the people who do the work.

Housing can go either way. Owning a home for yourself and your family would be far closer to personal property, while owning an apartment building to collect rent would be far closer to private property.

Socialism, for the most part and historically, is an umbrella term describing social rather than private ownership. That would include anarchism, which largely synonymous with "libertarian socialism." Lenin, on the other hand, used it to more specifically refer to an intermediate stage between capitalism in communism, so you might see people using that more narrow definition to exclude anarchists, democratic socialists, etc.