The definition of immigrant is merely a person who moves from one place to settle in another. Your definition is retsrictive and you have arbitrailly used it to support your point.
What more do you have to say? They are here against the laws of the US. That makes them illegal immigrant or aliens for that matter. What else do you need?
I want to keep the discussion within the bounds of the laws. Most people who visit America are called tourists. It makes no sense whatsoever to refer to those who come here without legal permission "immigrants".
I prefer calling them "tourists without visas".