I have heard that indio is used in Spanish to mean Indian, indigene, and Hindu.
Are all three meanings correct?
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The word "indio" referrs to someone from either the East Indies or the West Indies but contemporaneously usually refers to either an Indian national (someone from India or otherwise related to the Indian subcontinent) or someone ethnically related to the peoples who lived in the Americas before Columbus. I don't know about other parts of the Spanish-speaking world, but at least in Colombia "indio" is also a pejorative for someone deemed to be less educated or from a lesser cultural level (regardless of ethnic background).
The word "indígena" would roughly translate in English to native, aboriginal or indigenous, so it could be applied to a Lapp (a Same) in Finland, an Australian aboriginal, or to an indio from the Americas. It could also be extended to a white blonde person German in Germany, although this meaning is rare. Technically correct but rare. In the Americas, many people say "indígena" instead of "indio" to avoid the other two meanings of "indio" (someone from India or the insult). As with any other ethnically related word, some people adopt it with pride and some people will feel offended.
The word "hindú" correctly applies to either someone from historical Hindustan or a follower of Hinduism. By extension, and to avoid the use of "indio" (to avoid confusion with native Americans or the pejorative meaning) it is also applied to nationals of India. Most educated people will agree that "hindú" is incorrect when referring to just someone from India (unless related to Hindustan or Hinduism).
For avoiding confusion the word "hinduísta" refers to a follower of Hinduism regardless of ethnic or national identity.
Another related word is "amerindio" derived from "indio americano" which refers to someone indigenous to the Americas, avoiding any confusion with "indio" (and being more precise than "indígena"). Also, while some people proudly adopt the exonym "amerindio" others may take it offensively.
So, in summary:
Although indio is often used as a polysemic word meaning Indian, indigene, or Hindu, in Spanish we have specific words for each of these:
Hence, if using indio to refer to one of the above three meanings one must ensure the context is unambiguous.
Note: using indio for indigene sounds to me like some kind of insult. Indio is used sometimes in a pejorative way, when referring to south-americans.
I believe you are confusing the terms indio and hindú:
Indio is not to my knowledge used to refer to indigenous persons in general. The Spanish word for this is indígena or aborigen.
* Historically used to refer to persons from "Hindustan" (historical India, the Indian subcontinent): India, Pakistan, Bangladesh, Nepal, Sri Lanka, the Maldives, Bhutan.