trainer vs coach

trainer and coach both are nouns.

Word	Noun	Adjective	Verb	Adverb
trainer	Yes	No	No	No
coach	Yes	No	Yes	No

As nouns, coach is a hyponym of trainer; that is, coach is a word with a more specific, narrower meaning than trainer:

Other hyponyms of trainer include handler, manager, animal trainer, handler.

trainer (noun)	coach (noun)
simulator consisting of a machine on the ground that simulates the conditions of flying a plane	a carriage pulled by four horses with one driver
one who trains other persons or animals	a railcar where passengers ride
	a vehicle carrying many passengers; used for public transport
	a person who gives private instruction (as in singing, acting, etc.)
	(sports) someone in charge of training an athlete or a team

trainer (verb)	coach (verb)
	teach and supervise (someone); act as a trainer or coach (to), as in sports
	drive or operate a coach or carriage