Fair warning: It's their own fault if they get killed.

Economically the Peoples Republic of China is more what we would call "crony capitalism" than communist.

The difference is the actual communist part. The private sector is actually larger than the state owned sector and the state owned companies are shrinking. At best it is a mixed economy but they have pretty much given up on social programs and collectivism only keeping authoritarianism. I would say one party rule but in reality Xi is now dictator for life and everyone who opposed him is now either dead or in prison so all that is left is a rubber stamp.

Why recognise authoritarian regimes with state control of the economy as communist any more than authoritarian regimes with private control?

'private control" represents a significant source of eco-political power separate from the regime itself.

Because one can be authoritarian (Y axis) and be any where on the left right continuum (Y Axis). Left wing authoritarianism is pretty different from right wing authoritarianism while both are different with what is in between.

And what has that to do with anything? Communism is supposed to be a stateless society. Why does one perversion of the idea deserve the name while the other doesn't?

So, since North Korea calls itself a democracy, they're not communist, right? Or do we recognize that just because a country calls itself a thing, doesn't necessarily mean that it's true?

Because it's not about accuracy, it's all about rigging the score to show "communism" is losing. That's also why you always hear about African and South American socialist states here, but rarely the European ones.