booster rocket vs rocket engine

booster rocket and rocket engine both are nouns.

Word	Noun	Adjective	Verb	Adverb
booster rocket	Yes	No	No	No
rocket engine	Yes	No	No	No

As nouns, rocket engine is a hypernym of booster rocket; that is, rocket engine is a word with a broader meaning than booster rocket:

booster rocket: the first stage of a multistage rocket
rocket engine: a jet engine containing its own propellant and driven by reaction propulsion

Other hypernyms of booster rocket include rocket.

booster rocket (noun)	rocket engine (noun)
the first stage of a multistage rocket	a jet engine containing its own propellant and driven by reaction propulsion