birth vs lying-in

birth and lying-in both are nouns.

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

As nouns, lying-in is a hyponym of birth; that is, lying-in is a word with a more specific, narrower meaning than birth:

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

birth (noun)	lying-in (noun)
the event of being born	concluding state of pregnancy; from the onset of contractions to the birth of a child
a baby born; an offspring
the process of giving birth
the kinship relation of an offspring to the parents
the time when something begins (especially life)

birth (verb)	lying-in (verb)
cause to be born