It depends on wether you're working with a conception of history or if you're working with present-day definitions of common american political parlance. Corporatism did not straight up mean privatization. Only it was a political project with the aim to promote and secure private sovereignty over each sector of the economy. A society organized as a body - corpus. That is what Corporatism meant. You have the head, you have the arms, and the legs. Each part autonomous, and sovereign over it's sector. Just as the fingers don't tell the brain what to do, the employees must not contest the employer's ownership of their labour. If one knew what to do with finance or production, one was a factory owner or a banker. These last elements were to be empowered and secured by the Fascist state, an entity born entirely to subsume class struggle into nationalist fervour. Hence Corporatism.

The reason why Corporatism was not defined by the interdependence of State and Private Corporations is because it couldn't have been. As that would have been no different from Liberal or Socialistic States. All Industrialized economies are predicated on the state, on it's monopoly of violence, on it's ability to enforce property, contracts, to secure the money supply, and to galvanize social economic efforts. This even moreso true in the time period, as all the Liberal states of Europe were predicated on colonialism, just as the United States was predicated on manifest destiny. There's no capitalism or industrialization without state action. Corporatism's innovation - which is really an echo of early modern political thinking - was in ideologically subjugating civil society to the wishes of the private.

Nowadays people are too quickly to call any state they dislike 'corporatistic', 'corporativistic', or something along those lines because, ultimately, we live in society that is organized around those ideas. Every country from the liberal west to the post colonial east at one point played along Corporatist rules. Even when those Corporatist states were dismantled - early in the US, after the 80s oil shocks elsewhere - we continued to conceptualize the economy and our roles within it according to what the fascists had in mind. So the word loses all meaning. People will say the US is corporatist now because the State intervenes in the economy. Except it always did, and the hopes that the government would stop caring about social harmony and welfare, and limited itself to enforcing contract and property only, was always there. We merely live in a post Soviet acceleration.

Correct. Because the Nazi economy was called 'privatization.'