headsman vs public executioner

headsman and public executioner both are nouns.

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

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

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

Other hypernyms of headsman include executioner.

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