lying-in vs birthing

lying-in and birthing both are nouns.

Word	Noun	Adjective	Verb	Adverb
lying-in	Yes	No	No	No
birthing	Yes	No	No	No

As nouns, birthing is a hypernym of lying-in; that is, birthing is a word with a broader meaning than lying-in:

lying-in: concluding state of pregnancy; from the onset of contractions to the birth of a child
birthing: the process of giving birth

Other hypernyms of lying-in include birth, giving birth, parturition.

lying-in (noun)	birthing (noun)
concluding state of pregnancy; from the onset of contractions to the birth of a child	the process of giving birth