Question: Do I count as an international student if I completed all of my high school in another country but was born in the U.S.? And can I get financial aid if my parents have a house here in the U.S. and pay taxes?
You say that you were born in the U.S., so I assume that you are an American citizen. Is that correct? If so, then you are eligible for U.S. Federal financial aid, regardless of where you grew up and attended high school.
BUT ... beyond that, colleges will vary on how they regard you. Those that have special forms for international students or test requirements (e.g., TOEFL) for candidates whose first language isn't English (or whose high school instruction is not in English) may expect you to follow the protocol for international applicants. It makes sense for you to contact each admission office and ask where you fall, once you've finalized your college list. (This can also be a good way to connect with the staff member who oversees applicants from ... well ... overseas. ;))
However, financial aid is the biggie, and it sounds like you'll be fine in that department, no matter where you live.