Socialism is when the workers themselves possess the means of production. Your definition is not totally wrong if describing Soviet style governments.
Capitalist governments tell industries what to build fairly often, especially during war time. That doesn’t make the United States, Germany or any other country not capitalist because of that. Their systems still operate according to the law of value and uphold the institution of private property.
In fact, the Nazis began the trend of privatization long before the neoliberals such as Reagan and Thatcher began doing it. This was a unique policy because amid the Great Depression, most governments around the world were becoming more involved in the economy to promote economic stability and recovery.
Initially, the Nazi Party did not have “socialist” in its name, and it was added around 1920 to the objection of Hitler. Hitler would later clarify: “our adopted term ‘socialist’ has nothing to do with Marxian socialism. Marxism is anti-property; true socialism is not.” I implore you to find an online copy of Mein Kampf and search for “Marx” or “communism” and see that every time Hitler talks about them he describes them as Jew-dominated and revolting.