I am curious if others use the term this way.
I have not encountered people from outside the USA who use the term in this fashion.
If I referred to someone as "East Indian," would you assume that the
person was from India, from the eastern part of India, from somewhere
like Indonesia -- or would you just be confused (like me) and ask for
I would try to guess where the speaker was from, and would try to guess the context. In India, 'East Indian' is not commonly used. One would say "A person from Eastern India" to denote a person from that part. Since states of Eastern India do not have much in common, except poverty,and the language, culture and belief systems are pretty diverse, saying "A person from Eastern India" is not very useful, though. One can say Eastern India when referring to the region.
In India, there may be a handful of corporatiosn calling themselves East India/Indian xyz, to denote their place of origin (usually the state of West Bengal).
Saying "East India" in India may imply a reference to the British East India Company, which ruled (parts of) India for nearly a century, until 1857. Older people might think of Indonesia, which was called the Dutch East Indies.
By contrast, in India, West Indian almost universally refers to the Caribbean. India has strong cultural links with the Caribbean, involving cricket and migrants, among others. Again, people would use "A person from Western India" to refer to someone from western states.
However, North Indian and South Indian are very commonly used to describe people from the Northern and Southern part of India. The people from Eastern and Western India usually prefer to identify themselves with their state or language (Eg. Gujarati, Odia(Oriya), etc) rather than their geographical region.
I think that the usage of Indian, West Indian, and East Indian would be exactly the same in the whole of the commonwealth, except Canada, for obvious (historical) reasons.
Is there another, better way of saying "Indian from India"?
In most parts of the world, 'Indian' would suffice. 'South Asian' or 'Asian' is used in the UK and some other parts of the world to refer to people from the Indian Subcontinent; the former term is clearly more precise. In North and South America, especially in those parts where there is significant native population, one could say 'Asian Indian' and 'Native American' to distinguish between the two, and to avoid confusion. Indian from India is redundant and silly.
With time, one may expect that even in the USA, Indian would refer to people from India, while people would increasingly use more politically correct and accurate terms to refer to natives.