hair coloring vs dye

hair coloring and dye both are nouns.

Word	Noun	Adjective	Verb	Adverb
hair coloring	Yes	No	No	No
dye	Yes	No	Yes	No

As nouns, dye is a hypernym of hair coloring; that is, dye is a word with a broader meaning than hair coloring:

hair coloring: a dye for coloring the hair
dye: a usually soluble substance for staining or coloring e.g. fabrics or hair

Other hypernyms of hair coloring include dyestuff.

hair coloring (noun)	dye (noun)
the act of dyeing or tinting one's hair	a usually soluble substance for staining or coloring e.g. fabrics or hair
coloring of the hair
a dye for coloring the hair

hair coloring (verb)	dye (verb)
	color with dye
	color for microscopic study