public executioner vs slayer

public executioner and slayer both are nouns.

Word	Noun	Adjective	Verb	Adverb
public executioner	Yes	No	No	No
slayer	Yes	No	No	No

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

public executioner: an official who inflicts capital punishment in pursuit of a warrant
slayer: someone who causes the death of a person or animal

Other hypernyms of public executioner include killer.

public executioner (noun)	slayer (noun)
an official who inflicts capital punishment in pursuit of a warrant	someone who causes the death of a person or animal