hangman vs public executioner

hangman and public executioner both are nouns.

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

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

hangman: an executioner who hangs the condemned person
public executioner: an official who inflicts capital punishment in pursuit of a warrant

Other hypernyms of hangman include executioner.

hangman (noun)	public executioner (noun)
an executioner who hangs the condemned person	an official who inflicts capital punishment in pursuit of a warrant