Is earnings of American Express Bank’s branch located in Chennai included/excluded domestic income of India?

Yes,

It will be included in domestic factor income as the Bank’s branch is located within the domestic territory of India.

Explanation:-

The profits earned by the residents and non-residents (foreign companies) within the domestic territory of India is included in the domestic income.

All incomes and profits generated by all within the domestic territory of India will be included in the domestic income of India.