Lord's Day vs day of rest

Lord's Day and day of rest both are nouns.

Word	Noun	Adjective	Verb	Adverb
Lord's Day	Yes	No	No	No
day of rest	Yes	No	No	No

As nouns, day of rest is a hypernym of Lord's Day; that is, day of rest is a word with a broader meaning than Lord's Day:

Lord's Day: first day of the week; observed as a day of rest and worship by most Christians
day of rest: a day set aside for rest

Other hypernyms of Lord's Day include rest day.

Lord's Day (noun)	day of rest (noun)
first day of the week; observed as a day of rest and worship by most Christians	a day set aside for rest