The Japanese Manga in Chinese is called 漫画 (man4 hua4), the same way the Japanese called. Just that it is pronounced in Mandarin only.
The Japanese Anime in Chinese is called 動畫(动画) (dong4 hua4). Anime is not equal to manga, so 動畫 is not equal to 漫画.
Hence a Chinese cartoon is called 動畫.
動畫 in Chinese literally means motion pictures. In English term, we called it animation. Hence, the origin of the word "anime".
The word 漫画 is actually a Chinese-inspired-Japanese-made-word, and re-imported back to the Chinese language community again. It was based on the Chinese word 漫筆 (man4 bi3), literally means random writing, because it was used to describe drawings, so the Japanese changed the word 筆 (pen) to 画 (drawing), hence producing the term 漫画.