> One is : a society that is not capitalist, ie, the means of production are owned by the workers.

Among living socialists and communists, I'd argue that that is actually their definition of "socialism", while their definition of "communism" is a moneyless, classless and stateless society, the idea being that the former could lead to the later if implemented. All three words have changed their meanings over the last 200 years though. To most people today, "communism" just means an authoritarian state ruled by a communist party with a centralized economy and "socialism" means widely different things to different people depending on where you live and what media you read.