As an international applicant, loads of admission pages ask for an IELTs or TOEFL score to be submitted, but then add confusing bits after, like, “unless your first language is English” or “if you aren’t confident in your English”. How do you even quantify that? I’m perfectly fluent in English, but my country isn’t a generally English speaking country, so I don’t know whether I have to prove something by document. Can’t I just prove I’m fluent by sending in a good SAT score? Do I even need to prove that I’m fluent?
Ask the admission. Each school has different policies.
However, good number of school waive TOEFL requirements if you have certain SAT Reading score. That also depends on school
All universities will want to verify your base competency in the English language one way or another before offering you admission. There’s several good reasons to do that. One, it’s a requirement for the student visa, so might just as well get that out of the way early. Second, it drains resources from the university to support students who are struggling with the language and it’s just not a good experience for anyone involved.
By far the easiest way to document your English language proficiency is to take the TOEFL. It’s also the only form of documentation that’s universally accepted. Some (but not all) universities will waive their TOEFL requirement for sufficiently high scores on the SAT, or for students who have attended high schools where English was the primary language of instruction. However, some universities are firm on TOEFL scores being a pre-requisite for admission. You’d have to scratch those off your list if you opt not to take it.
Ha, I’d like to see that stated on an actual college website! 
Yeah, you’ll have to ask each school directly. Most schools will take a CR score above a certain threshold (usually in the mid-600s) in its place, but not all. For example, if I were to apply to CMU, I would have to submit a TOEFL even though I have lived in an English-speaking country for 15 years because my first language is technically not English (which I find quite ridiculous, but I digress).