eye doctor vs specialist

eye doctor and specialist both are nouns.

Word	Noun	Adjective	Verb	Adverb
eye doctor	Yes	No	No	No
specialist	Yes	No	No	No

As nouns, specialist is a hypernym of eye doctor; that is, specialist is a word with a broader meaning than eye doctor:

eye doctor: a medical doctor specializing in the diagnosis and treatment of diseases of the eye
specialist: practices one branch of medicine

Other hypernyms of eye doctor include medical specialist.

eye doctor (noun)	specialist (noun)
a medical doctor specializing in the diagnosis and treatment of diseases of the eye	an expert who is devoted to one occupation or branch of learning
	practices one branch of medicine