ok, i can understand this. so it’s small-business are private run while bigger businesses are organized differently?

Very small businesses would tend to be private. At least that's the trend. But keep in mind even a worker-owned co-op is a private business so it's one of those things where the exact implementation has varied. And many businesses that would otherwise be private and might be partially nationalized, requiring certain standards, barriers to creating a storefront, strict rules on employment (or not). It's something that countries can experiment with. Struggle doesn't end after revolution and the development of productive forces changes employment.

The best targets for nationalization are the big industries, the developed ones. They are much easier to plan and run and expropriate. But a country could do so earlier if conditions allowed. Keep in mind that every country with a socialist revolution has faced constant attack, so they have been under fairly desperate conditions to industrialize or otherwise abandon more idealistic forms of work, e.g. being the primary target of Nazis.

to sum it up, i’d like to add a point. businesses spend money mostly on 4 categories:

raw materials wages for workers machinery (this one has to be largely paid up-front) profits to shareholders

In Marxist terms you can usually summarize it in two: material inputs (dead labor) and labor. Profit isn't an input to production, it is actually variable and is just what is left over after all expenses. Income vs. expenditures. Raw materials and machinery are just commodities you buy to do production.

so to “socialize much of the gains” means that company profits are either redirected to the community (“state-run enterprise”)

It could be business taxation or it could be a redirecting of productive capacity (provide X prefab concrete widgets to this state project). Profits don't need to be redirected, that's just an accounting question. They may be made unprofitable by a given expropriation. But in countries taking this approach, like China, it's mostly business taxes.

or redirected into extra wages (“co-ops”)

If the co-op runs the business and is functional then the workers are basically setting their own wages and trying to optimize their business to pay as much as possible while still doing well as a business. If this is basically private enterprise then over time the state will begin direct involvement as it grows in size, observing production, expenditures, living conditions, etc. If a state allows small enterprise run more like a capitalist business, then eventually the state may step in a force cooperative governance, ensures a state-meditated union, etc.

But really this is exactly something that countries have experimented with, it's not just one prescription. And it has varied based on conditions, particularly external ones from imperialist countries.