public school vs middle school

public school and middle school both are nouns.

Word	Noun	Adjective	Verb	Adverb
public school	Yes	No	No	No
middle school	Yes	No	No	No

As nouns, middle school is a hypernym of public school; that is, middle school is a word with a broader meaning than public school:

public school: private independent secondary school in Great Britain supported by endowment and tuition
middle school: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of public school include Gymnasium, lycee, lyceum, secondary school.

public school (noun)	middle school (noun)
a tuition free school in the United States supported by taxes and controlled by a school board	a school for students intermediate between elementary school and college; usually grades 9 to 12
private independent secondary school in Great Britain supported by endowment and tuition