No Franco, you're incorrect. Socialism is the process that leads to high communism, according to Marx and Engels. The abolition of private property and markets, with an economy that is rationally and centrally planned. Communism is a stateless society, without socioeconomic classes or the need for money. In socialism, we still have the state and perhaps money as well.
USSR - UNION OF SOVIET SOCIALIST REPUBLICS
Where is the word "communist" there?
Communism is the end of the highest form of socialism
(the socialization and democratization of all production). True socialists have no problem, identifying themselves as communists, because communism is the end or objective of socialism. We identify ourselves with the objective.
In both socialism and in its goal, communism, all private property is defined as property that can be used to produce goods and services. It's essentially the means of production or facilities, lands, and machinery of production. Production at a national scale is a social endeavor, not a private one. It's fully democratized and publicly owned and run to meet human needs, rather than human greed, as what we have under capitalism and its markets. All production in socialism is non-profit.
Personal property is allowed in socialism and in communism. So one's home, vehicle, computer, and toothbrush. etc is your personal property and belongs to you.
So-called "market socialism" is an aberration, and both Marx and Engels clearly stated that markets and capitalists can exist in socialism, only in the interim, before production is fully socialized and democratized, in the hands of the public. Marx and Engels didn't like capitalism or markets, hence are very critical, barely only tolerating it in the beginning of the transition from capitalism to socialism. There might be a brief time when markets and capitalism are present, but it's not considered part of socialism. Capitalist-for-profit production, with products sold in a market, isn't socialism, which is defined as non-profit, publicly owned, centrally planned, and democratically-run production.
You're confusing a mixed economy
(capitalism with a little bit of socialism), with an actual socialist economy. As advanced automation continues to develop, socialism and later high-communism
(high-tech communism), inevitably become a necessity.