house detective vs private eye

house detective and private eye both are nouns.

Word	Noun	Adjective	Verb	Adverb
house detective	Yes	No	No	No
private eye	Yes	No	No	No

As nouns, private eye is a hypernym of house detective; that is, private eye is a word with a broader meaning than house detective:

house detective: a private detective employed by a hotel or retail store
private eye: someone who can be employed as a detective to collect information and assist in criminal investigations

Other hypernyms of house detective include operative, PI, private detective, private investigator, shamus, sherlock.

house detective (noun)	private eye (noun)
a private detective employed by a hotel or retail store	someone who can be employed as a detective to collect information and assist in criminal investigations