private instructor vs teacher

private instructor and teacher both are nouns.

Word	Noun	Adjective	Verb	Adverb
private instructor	Yes	No	No	No
teacher	Yes	No	No	No

As nouns, teacher is a hypernym of private instructor; that is, teacher is a word with a broader meaning than private instructor:

private instructor: a person who gives private instruction (as in singing, acting, etc.)
teacher: a person whose occupation is teaching

Other hypernyms of private instructor include instructor.

private instructor (noun)	teacher (noun)
a person who gives private instruction (as in singing, acting, etc.)	a personified abstraction that teaches
	a person whose occupation is teaching