'Companies', 'hiring, 'workers' are shorthand for common practices that come after. Like how a 1-3-5 is a common structure for music notes that it has become its own thing.

Both your definitions are correct, but you have a misunderstanding that is resulting in you making the incorrect extension to the meaning of socialism:

>any given company is owned by the people that are working for the company

Companies are an invention given a name because of common practice. The definitions of socialism and capitalism don't allude to companies, only people. So a corrected version would be:

>any given tools are owned by the government of the people working with the tools

A company is shorthand for when a person hires 'private contractors', those contractors buy or borrow tools from the person, and the person borrows tools/worker time but only with as much leverage and assets are included in their work and not things they use to survive.

I'm still simplifying a bit, but it's a complex series of interactions that has been shorthanded for convenience to 'a company'.

So why is that distinction important?

Because a person hiring private contractors and paying them a proportion of total profit is still capitalism. If the private contractors buy the tools or are even given them to own upon starting their help... still capitalism. It's just different versions of how a person can buy help.

Why does that distinction get blurred?

Because we now have people who have unified so much of a particular production they practically have the control of a government. And viola you mistakenly swapped government for company in your definition of socialism. Capitalism assumes anarchy and greed (an imperfect assumption), so having huge companies is anti-capitalist because it's too ordered and collaborative (the feudalists argument is to justify their greed because of capitalism even though they are no longer operating in a good capitalist environment).

So there are 2 options. You take the more capitalist approach: break up large companies and force a greater degree of anarchy. Or you recognize that large companies are essentially a government and take the socialist approach: no one may own the company or its assets and it is instead run like a government (democracy being our default). In other words the capitalist profit sharing method of hiring would become the mandatory method of hiring, so companies will naturally inflate into democracies instead of monarchies. Both options have a laundry list of technical struggles that mostly stem from the same issues the capitalist model struggles with: people being coercable, and people being rational (knowing economics and transparency to use that knowledge so caveat emptor is also anti-capitalist)