While I cannot provide you with information on how history was studies in India or China, I can tell you that what you are looking for is called "historiography".

Historiography is basically about the following questions:

• how did the discipline of history develop
• how do and did historians work
• how was a specific topic studied and portrait in different times

Unfortunately I am not very familiar with the field of historiography, so I am unable to recommend you some readings from first hand, but one of my professors highly praised the book "Discovering History in China: American Historical Writing on the Recent Chinese Past" by Paul A. Cohen. Cohen is looking at how the history of China was studied in the West, particular in America.

I am sure some other redditors can expand on this and provide you with readings.