If you ask a dozen pregnant women “who is going to deliver your baby?” their answers will likely be the names of their midwives, doctors or medical group. It is rare for a woman to say “I am!” The reality is that while we might say that someone can deliver another person’s child, only mothers give birth to their babies.

https://www.plumtreebaby.com/blogs/news/17956548-deliver-vs-give-birth
Здесь больше говорится о глубинном смысле этих выражений, но, насколько я знаю, deliver a baby употребляется в значении родить ребенка. А так, конечно, я согласна, что give birth несет в себе истинный смысл.

deliver a baby means to help a pregnant woman give birth

+++Lena

there are many examples of using "deliver a baby" as a synonym to "give birth"...(?)
https://books.google.com/ngrams/graph?content=women+delivering+babies%2C+women+delivered+babies&year_start=1990&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cwomen%20delivering%20babies%3B%2Cc0%3B.t1%3B%2Cwomen%20delivered%20babies%3B%2Cc0

Ok, Elena 🙄

The women in those examples delivered babies that other women gave birth to.

I'll delete it, no problem, it's just unclear to me

If you ask a dozen pregnant women “who is going to deliver your baby?” their answers will likely be the names of their midwives, doctors or medical group. It is rare for a woman to say “I am!” The reality is that while we might say that someone can deliver another person’s child, only mothers give birth to their babies.

Lena cleared it up before anybody.