You are not too far off with the above, but in reality Nazism is closer to Marxism than it is free market capitalism.
So the subtle difference is with Nazism the state took a slightly less direct control of seizing the means of production, but still maintained complete control nevertheless. Private property wasn’t outright abolished but it had to serve the interest of the state or else be subjected to seizure. Nazi party leaders served on the board of every major corporation.Article 153 of the Weimar constitution guaranteed private property, with expropriation only to occur within the due process of the law, but this article was nullified by a decree on February 28, 1933.
With this, the new National Socialist government had complete control of private property in Germany. While they did not take complete control of the lands like the Bolsheviks did in Russia in 1917, the Nazis issued quotas for industries and farms, and later they reorganized all industry into corporations run by members of the Nazi Party.
Both communists/socialists and fascists controlled every aspect of its citizens’ lives. Both prioritized the community over the individual.