hair coloring vs colouring

hair coloring and colouring both are nouns.

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

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

Other hypernyms of hair coloring include coloring.

hair coloring (noun)	colouring (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