running hand vs handwriting

running hand and handwriting both are nouns.

Word	Noun	Adjective	Verb	Adverb
running hand	Yes	No	No	No
handwriting	Yes	No	No	No

As nouns, handwriting is a hypernym of running hand; that is, handwriting is a word with a broader meaning than running hand:

running hand: rapid handwriting in which letters are set down in full and are cursively connected within words without lifting the writing implement from the paper
handwriting: something written by hand

Other hypernyms of running hand include hand, script.

running hand (noun)	handwriting (noun)
rapid handwriting in which letters are set down in full and are cursively connected within words without lifting the writing implement from the paper	the activity of writing by hand
	something written by hand