dance vs stage dancing

dance and stage dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
dance	Yes	No	Yes	No
stage dancing	Yes	No	No	No

As nouns, stage dancing is a hyponym of dance; that is, stage dancing is a word with a more specific, narrower meaning than dance:

dance (noun)	stage dancing (noun)
taking a series of rhythmical steps (and movements) in time to music	a show involving artistic dancing
an artistic form of nonverbal communication
a party for social dancing
a party of people assembled for dancing

dance (verb)	stage dancing (verb)
move in a pattern; usually to musical accompaniment; do or perform a dance
move in a graceful and rhythmical way
skip, leap, or move up and down or sideways