The term "higher education" is completely wrong. Education is not a strict total order, so one cannot speak of "higher" forms of education.
For instance, if I learn to speak French in school A, and if I learn to solve differential equations in school B, then which form of education is "higher"?
"higher education" is a term of art that refers to the same thing also referred to as postsecondary education, e.g., in US terms college/university education. Insofar as the relation to the common use of "higher" alone is inspired by a presumed total order, it is the chronological order in which various types of schools are attended in the normal course (in modern practice, this is less than perfectly strict.)
For instance, if I learn to speak French in school A, and if I learn to solve differential equations in school B, then which form of education is "higher"?