Actually, much of Chinese industry is privately owned. So that identifying characteristic makes them more capitalist than communist.
Chinese style capitalism is quite different than in the US.
In addition to the fully owned and operated state businesses in China, there are the state controlled and run businesses such as banking, telecom or transport which have minority private ownership.
Then the're the joint ventures between private (often foreign) companies and Chinese state-backed entities with the foreign firms providing the technology and its Chinese partner provides access to the Chinese market. Joint ventures are common in fields such as car making, logistics and agriculture.
Next is the fully private business. They are still subject to frequent meddling by the government. If they are favored, state-controlled banks will provide them with cheap loans and bureaucrats will attempt to discourage their foreign competitors. Such meddling is common in areas such as energy and the internet.
And finally there're are businesses fueled by investment by local government, often through municipally owned venture-capital funds. These funds typically back businesses that dabble in clean tech or hire locals. There are thousands of these small businesses in towns and villages across China.
The term private business in the US is pretty clear but not so in China. However, whatever they have seems to be working quite well for them.