hair coloring vs coloring

hair coloring and coloring both are nouns.

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

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

Other hypernyms of hair coloring include colouring.

hair coloring (noun)	coloring (noun)
the act of dyeing or tinting one's hair	the act or process of changing the color of something
coloring of the hair	a visual attribute of things that results from the light they emit or transmit or reflect
a dye for coloring the hair	a digestible substance used to give color to food