engagement vs call-back

engagement and call-back both are nouns.

Word	Noun	Adjective	Verb	Adverb
engagement	Yes	No	No	No
call-back	Yes	No	No	No

As nouns, call-back is a hyponym of engagement; that is, call-back is a word with a more specific, narrower meaning than engagement:

Other hyponyms of engagement include shape-up, booking, reservation.

engagement (noun)	call-back (noun)
the act of sharing in the activities of a group	the recall of an employee after a layoff
a hostile meeting of opposing military forces in the course of a war	a return call
the act of giving someone a job
employment for performers or performing groups that lasts for a limited period of time
a mutual promise to marry
contact by fitting together
a meeting arranged in advance