I live in a red state, it is common to call any collective action communism. It’s common historically for the right to call collective action or ownership communism.
Corporate socialism? Honestly never heard that. Market Socialism is the form of socialism that seeks to have publicly-traded companies abolished and replaced by co-ops. Mixed with state owned companies in the case of natural monopolies and basic services (transport, electricity, education, health, etc).
Market Socialism is very much aligned with OG Marxism, so I see how red-scared people may call it communism.