늦잠을 자다 meaning/connotation
In English to oversleep is bad. If you oversleep you woke up later than you wanted to and may be late for work. If you lie-in (or sleep in as I believe US English speakers say), it's something good. If you had a lie-in you didn't have to go to go anywhere or do anything, and can enjoy the extra sleep.

하지만 한국어로 똑 같은 말을 말하세요? (In Korean, do you use '늦잠을 자다' for both situations?)

그리고 'I went to bed/sleep late' 말하면 '늦잠을 자다'도 말하세요?

Question

늦잠을 자다  meaning/connotation
In English to oversleep is bad. If you oversleep you woke up later than you wanted to and may be late for work. If you lie-in (or sleep in as I believe US English speakers say), it's something good. If you had a lie-in you didn't have to go to go anywhere or do anything, and can enjoy the extra sleep.

하지만 한국어로 똑 같은 말을 말하세요? (In Korean, do you use '늦잠을 자다' for both situations?)

그리고 'I went to bed/sleep late' 말하면 '늦잠을 자다'도 말하세요?

Answer

Nope, the meaning of "늦잠을 자다" is same as "Oversleep" in English. 
I don't think there is a word that has same meaning with "lie-in" in Korean. (Or maybe I can't remember it)

And "늦잠을 자다" only represents "Oversleep". In Korean, "I went to bed/sleep late" is "늦게잤다"

Answer

늦잠을 자다 can be used at both situation when you oversleep and lie-in/ sleep in.
If you would like to underline that you "lie-in/ sleep-in," you can say "늦잠을 즐기다." It can be literally translated in "enjoy oversleeping"

늦게 잠들었다 / 늦게 잠자리에 들다
go to bed/sleep late