No, socialism means public ownership/control of the means of production. "Public" typically means state. That's how economists define it, that's how politicians define it, that's how all of the dictionaries define it. For example, England's NHS is socialism and was created by Clement Attlee, who was a socialist.
Merriam-Webster: any of various economic and political theories advocating collective or governmental ownership and administration of the means of production and distribution of goods
Oxford English A political and economic theory of social organization which advocates that the means of production, distribution, and exchange should be owned or regulated by the community as a whole.
American Heritage: Any of various theories or systems of social organization in which the means of producing and distributing goods is owned collectively or by a centralized government that often plans and controls the economy.
Collins: an economic theory or system in which the means of production, distribution, and exchange are owned by the community collectively, usually through the state.
Macmillan: a political system that aims to create a society in which everyone has equal opportunities and in which the most important industries are owned or controlled by the whole community
Cambridge: any economic or political system based on government ownership and control of important businesses and methods of production
Wiktionary: Any of various economic and political theories advocating collective or governmental ownership and administration of the means of production and distribution of goods.
Infoplease: general term for the political and economic theory that advocates a system of collective or government ownership and management of the means of production and distribution of goods.
Dictionary.com: a theory or system of social organization that advocates the vesting of the ownership and control of the means of production and distribution, of capital, land, etc., in the community as a whole.
Allwords: Any of various political philosophies that support social and economic equality, collective decision-making and public control of productive capital and natural resources, as advocated by socialists.
MnemonicDictionary: a political theory advocating state ownership of industry
FreeDictionary: Any of various theories or systems of social organization in which the means of producing and distributing goods is owned collectively or by a centralized government that often plans and controls the economy.
Socialism means public ownership/control of the means of production. It is not welfare statism, which is a different, disastrous idea from the left.
If a group of people own a company and are working for profits in a market economy, then they are capitalists.