The words typically used will vary by country and, within the US, by region. Your words would be used in the US, except that I think “elementary” is more common than “primary”. Some places use junior high school and senior high school, others use middle school and high school.
“Secondary” schooling isn’t a term that would colloquially be used in the US. It would be used by education professionals. It would include high school and possibly parts of middle school.