spit vs cough up

spit is a noun but cough up is not a noun.

Word	Noun	Adjective	Verb	Adverb
spit	Yes	No	Yes	No
cough up	No	No	Yes	No

As verbs, cough up is a hypernym of spit; that is, cough up is a word with a broader meaning than spit:

Other hypernyms of spit include cough out, expectorate, spit out, spit up.

spit (noun)	cough up (noun)
the act of spitting (forcefully expelling saliva)
a skewer for holding meat over a fire
a clear liquid secreted into the mouth by the salivary glands and mucous glands of the mouth; moistens the mouth and starts the digestion of starches
a narrow strip of land that juts out into the sea

spit (verb)	cough up (verb)
expel or eject (saliva or phlegm or sputum) from the mouth	discharge (phlegm or sputum) from the lungs and out of the mouth
utter with anger or contempt	give reluctantly
drive a skewer through
rain gently