There is no scientific defintion of race but I think you are confusing race with ethnicity.
By your definition Mexico has hundreds of races then.
Mexico is a multi-racial society. ;)
Mexico IS a multi racial society.
And, with respect to China and India, I am not confusing race and ethnicity, as you suggest. I am well aware of the difference.
You know from your common sense knowledge that race and ethnicity often parallel one another as breeding populations also transmit ethnic and cultural characteristics, so some confusion is not surprising,and you could reasonably think I might be confused on the subject. But, I am not.
I will point out that not only does India harbor different races, but the races of India are less closely related than are Europeans and black Africans, since both Africans and Europeans are representatives of "post-Toba" populations, but some southern Indian racial group(s) belong to "pre-Toba" populations. There aren't too many of these populations left anywhere on earth. You have the Andaman Islanders, some hill tribes in Malaysia and Indonesia, and some southern Indain groups and that's about it.
The distinction between pre and post Toba human populations is purely racial and derives from different genetic material. That these groups might have differing cultural or ethnic traits paralleling their racial ones is acknowledged, but that was not my point.