Communist Party of China follows the naming conventions used with other communist parties. Examples include: the Communist Party of the Soviet Union, The Communist Party of the United States of America, communist party of Cuba, communist party of Vietnam, The Communist Party of Portugal, etc. That naming convention is simple. You have a communist party located in a country/nation/geographic region/political entity.

Saying "Chinese Communist Party" is meant to place the emphasis on Chinese which in English has ethnic implications. It's the vision of the party as some kind of Han ethnonationalist movement (instead of a communist movement located in China). Thus, they're foreigners in Tibet, Mongolia, Xinjiang, and maybe even in Cantonese-speaking China. The history of China as a political entity is handwaved-away.

It's a subtle difference because "Of China" and "Chinese" can linguistically serve the same function in English. It's not a purely linguistic difference, it's mainly the baggage with how the terms have been used.

CPC is the official and correct translation. CCP is wrong and the western media use it because they have already demonised the term.

I might be wrong but i think it's in part to stay in line with the naming conventions of communist parties elsewhere while also emphasizing national unity through "Communist Party of China" whereas "Chinese communist party" makes it sound like it's a more ethnically oriented movement (lending itself to western propaganda about chinese immigrants by implying legitimacy over the Chinese, not China). Calling it the CCP helps let westerners get away with criticizing the entirety of China while claiming they're only criticizing the government. Also they're probably aware that Americans are morons and CCP is close enough to CCCP that they'll react with cold war

Its from the Theses presented to the second World Congress of the Communist International, it was the standard to my understanding for members of the ComIntern to adopt this convention:

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generally binding resolutions shall be passed only on such questions, on which such resolutions are possible.

18. In connection with the above all Parties desiring to join the Communist International must alter their names.

Each Party desirous of joining the Communist International must bear the following name:

Communist Party of such-and-such country (Section of Third Communist International). The question of denomination of a party is not only a formal one, but it is a political question of great importance. The Communist International has declared a decisive war against the whole bourgeois world and all yellow Social Democratic Parties. It is necessary that each rank-and-file worker should be able to distinguish clearly the difference between the Communist Parties and the old official "Social Democratic" or "Socialist" parties, which have betrayed the cause of the working class.

CCP carries an implication that the party practices a communism/socialism that is “Chinese” in ideology. It presents “Chinese Communism” as being a school of communism, a la Marxist-Leninism or Trotskyite, rather than a Communist party of a particular country.