I am a US citizen living in South India. Where do I have to apply for visiting China?

You can apply the visa in India, since you live in India ( usually with a valid resident permit).
The Consular Districts of Consulate-General of the People's Republic of China in Mumbai are Maharashtra State and Karnataka State while the Consular Districts of Consulate-General of the People's Republic of China in Kolkata are the State of Bihar, Chattisgarh, Jharkhand, Orissa and West Bengal. You should go to the related China Consulate to apply for a visit visa.