"Free Market" capitalism is just a buzzword in my opinion that does not denote anything other than the specter of regulation (or lack of) in our hyper-capitalistic world. It is used as a facade for capitalism and the resulting corportocracy to legitimatize their policies much like law and order is used to legitimatize liberalism despite equality being basically non-existent.
https://evonomics.com/why-free-market-ideology-is-a-double-lie/
You analysis of Italy and Germany fascism is suspect as they both had many industries and related commerce that were not controlled directly by the state. If you accept that it is not complete control your definition could be considered very broad because most nations have some control over their industries or, in the case of the US and other nations, the industries control the government through regulatory capture.
China, of all places, would fit your definition to a T, but I highly doubt they would appreciate being called fascist, even if they are.
You speak as if Capital struggles to weaken the state to capitalize (ie often creating monopolies). This is not the case at all as seen in the policies for modern countries for the last few hundred years. Corporations, not governments, have become the dominant social structure.
How are they connected? I think you may not be able to see this because you believe fascism can only exist in a specific economic model brought by a dictator's hand as opposed to a natural result of corporate control of government.
Essentially modern fascism is a corporatocracy regardless of how it is instituted and I would argue that this has evolved into neofascism and it's best friend neoauthoritarianism (especially in Asia, but now the US as well) we are experiencing currently.
When I jokingly said fascism = bad I was only half joking. Obviously most people operate on this simple definition. This is the framework that prevents us from accepting that fascism is currently dominating the planet because people reject being labeled as bad. This is why I think it is important to understand modern fascism not purely as an ideology that is wrong, but a valid ideology that is the most politically powerful and economically advantageous position the wealthy can be in.
https://tomorrowsaffairs.com/corporations-as-superpowerswelcome-to-the-world-of-altered-sovereignty
