Christopher Colombus has made a big mistake and I am confused when I am told there is an Indian in the US because how do I know that is a real Indian from India now working in Silicon Valley or a real Indian living in this continent for many many years and never leaving?
Usually, you can tell from the context of the conversation. Yes, the Cleveland Indians is a classic example of using "Indians" to mean "Native Americans". If I say, "We are playing cowboys and Indians", then you know which Indian it is. I hear Indians all the time in the U.S. when referring to Native Americans, but the person usually says "American Indians" except when talking about the Cleveland Indians. In fact, I'm wearing a Cleveland Indians shirt right now!
"Native American" might be a good term to use. But be aware that almost any name could be construed as racist or insensitive. You can read more at this Wikipedia article.
There is an ambiguity in words. If someone says "American" does one include everyone from the continents of North and South America. Does "Indian" include the whole Indian subcontinent, or only the modern state of India.
"Indian" is used for Native Americans, and people from (or descended from) India or the Indian subcontinent. This is played for laughs in The Simpsons
Apu: I am no longer an Indian living in America; I am an Indian American.
Lisa: You know what, Apu, in a way, all Americans are immigrants; except Native Americans.
Homer: Yeah, native Americans like us.
Lisa: No I mean American Indians.
It is normally not necessary to talk about someone's race. On those rare occasions when it does become necessary you should use terms like "Native American", or "First Nation" in Canada. If someone uses the term "Indian", it is probably clear from context which meaning is meant, or it doesn't matter.
Give this paper to Kat
She's the Indian woman in the end cubical.
There is an ambiguity, but it doesn't matter. In fact the last speaker should have just said "She's in the last end cubical", as it wasn't necessary to mention Kat's ethnicity.
There may be very rare situations in which precision is needed. In which case you can ask for clarification. But note the warnings: It is normally not necessary to discuss someone's race, and the ambiguity, if it arises, often doesn't matter.