

That’s not true at all
Socialism - any system of government where the workers own the means of production. This has nothing to do with welfare or healthcare or any of that, just that the working class owns the factories and other private property. (The marxist and original definition of private property is property that generates capital, not your toothbrush, which is personal property. Same with your home.)
communism - a post socialist system where the workers own the means of production, currency has been abolished, there are no class divisions or state (note the marxist concept of the state is probably not what you’re thinking if this is new to you.)



yes, but none of them use your definitions, those are colloquial ones not used by any communist philosophers.