I think the difference is when you are told to build an outbuilding, and told what you will store in it. In the South, some slaves ran businesses for their masters, but they still didn't have ultimate control of what they built and ran.
Two really good examples.
In the first place, the government tells various businessmen to build things for them -- space ships, for example -- and those companies go and build it to the government specifications, usually at an agreedupon price.
In the second place, even a business that is CONTROLLED by slaves is still OWNED by the slaves' masters whether he actually does anything or not.
A government may exert control of, say, the banking industry or the finance sector by imposing regulations or by sharply restricting activities it considers damaging to the economy. It may exercise control of oil companies by restricting where they can and cannot drill for new oil sources, or it can require them by force of law to pay to repair damage caused by their occasional mishaps.
This becomes socialism if and only if the government takes direct OWNERSHIP of the company away from its shareholders or other proprietors. It would still BE socialism even if the state-owned companies were run on a for-profit basis as if they were private companies, even if they were amazingly profitable, even if the government ministers who nominally supervised them made no attempt whatsoever to exercise control over the appointed CEOs.
I fully concede that there's sometimes a very thin line between social democracy and outright socialism. It must be said, however, that that thin line is the difference between private and government ownership of the means of production
Ownership is a problematic measure because under many systems, such as most of the third world, it can't even be accurately determined, with overlapping claims that depend more on exerting will or ignoring what the official paperwork says.
Do not confuse theft or corruption with an actual dispute of ownership. It is common practice in some Latin American countries to disenfranchise certain groups (certain ETHNIC groups, although Latinos on both sides are loathe to admit this) through legal and logistical tricks that may or may not even be legal. In some cases it's a bit like the Predatory Lending scandals in the U.S., but on a much larger scale because their governments are a lot less responsive to those kinds of problems.