The letter addressed 3 distinct persons. Citizens, illegals and immigrants. Why do I say this? Between the word illegal and immigrant is the word "or". So this 3 type of persons are distinct and not the same.
The letter is indeed addressed to three types of persons.
However, only two of those types of persons are mutually exclusive and therefore "distinct".
According to Webster's Dictionary, an immigrant is;
a person who comes to a country to take up permanent residence
"Immigrant" has no bearing on your current citizenship status but only has a bearing on your past place of birth and how you ended up in the United States of America.
If you fell out of your mother's birth canal and landed in the United States of America, you are a native-born citizen and, in accordance with Article II, Section 1 of the Constitution, are eligible to be President of the United States.
If you are a person who came to the United States of America to take up permanent residence, you are and will always be an immigrant. You may later become a citizen but your immigrant status will always disqualify you from ever being President of the United States in accordance with Article II, Section 1 of the Constitution.
Of the three "types of persons" you mention, the adjectives deal with place of birth and citizenship status that are separate legal issues and can yield several different combinations of legal status.
Immigrant + illegal entry = Illegal Alien not eligible to vote
Immigrant + legal entry = Resident Alien not eligible to vote
Immigrant + legal entry or amnesty + naturalization = U.S. citizen eligible to vote after age 18 but not eligible to be President of the United States
Native born = U.S. citizen eligible to vote after age 18 and eligible to be President of the United States
The only "types of persons" that are "distinct" and mutually exclusive are "native born" and "illegal".
The Immigrant, however, can be anything from an illegal alien not eligible to vote to a legal resident not eligible to vote to a U.S. Senator or a Governor of California eligible to vote but not eligible to become President of the United States.