Some Teachers even at public schools are not considered to be government employees but in reality since the teachers salaries are being paid for by the government taxes then I would personally consider all teachers who work at public schools to be employees of the government.
Anytime the taxes from government pays someone's salaries you're technically a government employee and are working for the public.
So yes teachers as long as they're working at public schools and are paid their salaries by the taxes people pay then they're government employees and work for the public.
So they're also public employees employed by the public who pay their salaries.
Teachers at private Schools though are paid directly from the parents so the teachers at private schools are not government or public employees.