A society where public ownership of the means of production, a state controlled by a politically organized proletariat, and production for societal use rather than for profit is the principal aspect (main body) of the economy.

Key term here is principal aspect. There is a weird phenomenon from both anti-communists as well as a lot of ultraleft and leftcom communists themselves of applying a “one drop rule” to socialism, where socialism is only socialism if it’s absolutely pure without a single internal contradiction. But no society in the history of humankind has been pure, they all contain internal contradictions and internal contradictions are necessary for one form of society to develop into the next.

If you applied that same logic to capitalism, then if there was any economic planning or public ownership, then capitalism would cease to be “true capitalism” and become “actually socialism”, which is an argument a lot of right-wing libertarians unironically make. The whole “not true capitalism” and “not true socialism” arguments are two sides of the same coin, that is, people weirdly applying an absolute purity standard to a particular economic system which is fundamentally impossible to exist in reality, so they then can declare their preferred system “has never truly been tried”. But it will never be tried ever because it’s an idealized form which cannot exist in concrete reality, actually-existing capitalism and socialism will always have internal contradictions within itself.

If no idealized form exists and all things contain internal contradictions within themselves, then the only way to define them in a consistent way is not to define them in terms of perfectly and purely matching up to that idealized form, but that description merely becoming the principal aspect in a society filled with other forms and internal contradictions within itself.

A capitalist society introducing some economic planning and public ownership doesn’t make it socialist because the principal aspect is still bourgeois rule and production for profit. This would mean the state and institutions carrying out the economic planning would be most influenced by the bourgeoisie and not by the working class, i.e. they would still behave somewhat privately, the “public ownership” would really be bourgeois ownership and the economic planning would be for the benefit of the bourgeoisie first and foremost.

A similar story in a socialist society with markets and private ownership. If you have a society dominated by public ownership and someone decides to open a shop, where do they get the land, the raw materials, permission for that shop, etc? If they get everything from the public sector, then they exist purely by the explicit approval by the public sector, they don’t have real autonomy. The business may be internally run privately but would be forced to fit into the public plan due to everything around them demanding it for their survival.

Whatever is the dominant aspect of society will shape the subordinated forms. You have to understand societies as all containing internal contradictions and seeking for what is the dominant form in that society that shapes subordinated forms, rather than through an abstract and impossible to realize idealized version of “true socialism”.

Countries like Norway may have things that seemingly contradict capitalism like large social safety nets for workers funded by large amounts of public ownership, but these came as concessions due to the proximity of Nordic countries to the USSR which pressured the bourgeoisie to make concessions with the working class. However, the working class and public ownership and economic planning never became the principal aspect of Norway. The bourgeoisie still remains in control, arguably with a weaker position, but they are still by principal aspect, and in many Nordic countries ever since the dissolution of the USSR, the bourgeoisie has been using that dominant position to roll back concessions.

The argument for China being socialist is not that China has fully achieved some pure, idealized form of socialism, but that China is a DOTP where public ownership alongside the CPC’s Five-Year plans remain the principal aspect of the economy and other economic organization is a subordinated form.

Deng Xiaoping Theory is not a rejection of the economic system the Soviets were trying to build but a criticism of the Soviet understanding socialist development. After the Soviets deemed they had sufficient productive forces to transition into socialism, they attempted to transition into a nearly pure socialist society within a very short amount of time, and then declared socialist construction was completed and the next step was to transition towards communism.

Deng Xiaoping Theory instead argues that socialism itself has to be broken up into development stages a bit like how capitalism also has a “lower” and “higher” phase, so does socialism. The initial stage is to the “primary stage” of underdeveloped socialism, and then the main goal of the communist party is to build towards the developed stage of socialism. The CPC disagreed that the Soviets had actually completed their socialist construction and trying to then build towards communism was rushing things far faster than what the level of productive forces of the country could sustain and inevitably would lead to such great internal contradictions in the economic system to halt economic development.

The argument was not a rejection of the Marxist or Marxist-Leninist understanding of what socialism is, but a disagreement over the development stages, viewing socialism’s development as much more gradual and a country may remain in the primary stage like China is currently in for a long, long time, Deng Xiaoping speculated even 100 years.

I recall reading somethings from Mao where he criticized the Marxian understanding of communism, but not from the basis of it being wrong, but it being speculative. He made the argument that Marx’s detailed analysis of capitalism was only possible because Marx lived in a capitalist society and could see and research its development in real time, therefore Mao was skeptical the current understanding of communism would remain forever, because when you actually try to construct it you would inevitably learn far more than you could speculate about in the future, have a much more detailed understanding of what it is in concrete reality and what its development stages look like.

In a sense, that’s the same position the modern CPC takes towards socialism, that the Soviets and Mao rushed into socialism due to geopolitical circumstances and did not have time to actually fully grasp what socialist development would look like in practice, and Deng Xiaoping Theory introduces the concept of the primary stage of socialism based on their experience actually trying to implement it under Mao.

Despite common misconception, the CPC’s position is indeed that China is currently socialist, not “will be socialist in 2049” or whatever. The argument is that China is in the primary stage of socialism, a system where socialist aspects of the political and economic system have become the main body but in a very underdeveloped form.