For...clarification, not that it matters.
Communism is in the name, communal, community. A business that is "worker owned" is communist. Though in a Marxist vision, it first has to be taken over by the "dictatorship of the proletariat", a community-oriented dictatorship that will take from the "owners" and give it over to the community, the people. Communisim has some regional variations on this, but it's largely the same thing. Proletariat rebels against the bourgeoisie, seizes the means of production and control of the government, uses the government to repurpose the means of production to producing for the people and not for profit.
Socialism has variations, America has practiced "soft socialism" for a long time. Roads, military, Amtrak, licensing, public education, and others. Total ownership of certain industries because their benefit to the nation is too profound to let them be operated privately and for profit. There are many areas where public services mimic businesses (why do schools need boards of administrators? Why do hospitals needs boards of trustees?) Other industries the government subsidizes and and heavily regulates because their proper operation is too important to the national good to do otherwise, but they are not so incredibly important that they can't also be privately run.
Most European countries practice a stronger form of socialism, actively requiring labor leaders and government officials to be on the boards of companies and requiring some portion of the business to be owned by the workers and/or the government.
And yes it goes all the way to hard socialism, which is a hair shy of communism, except that businesses can be privately owned, but are effectively under the complete control of the government. If they want you to bake blue cakes instead of red cakes you do that now, or you're not allowed to own a shop anymore. People can still earn profit, do jobs, but these things are all very heavily regulated.
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Subsidies are capitalists bullshitting you into thinking socialism isn't socialism. We're taking money from XYZ and giving it to ABC. That's socialism. The primary purpose of socialism is to redistribute from the haves to the have nots, money, power, control, influence, all of that. Subsidies as America does them are 99.9% of the time, the reverse, taking from the have nots (the average taxpayer) and giving to the whiny haves (wealthy landowners) because they've abused the capitalist model to such a degree that they can't even capitalist anymore. The end result is usually massive government bailouts followed by bankruptcies where the guy who just got all that taxpayer money runs off to the Bahamas for the rest of their life. Because the government thinks it can buy compliance and is absolutely terrified of executing its rightful power to tell people to get all their shit together and run their business properly.
Capitalism isn't a problem, people are just dicks.