According to Online Etymology Dictionary, baby comes from babe:
late 14c., short for baban (early 13c.), which probably is imitative
of baby talk (see babble), however in many languages the cognate word
means "old woman" (compare Russian babushka "grandmother," from baba
Crist crid in cradil, "moder, baba!" [John Audelay,
Now mostly superseded by its diminutive form baby. Used
figuratively for "a childish person" from 1520s. Meaning "attractive
young woman" is 1915, college slang. Babe in the woods is from 1795.
The word fetus is rather a medical term like embryo. Its usage picked up sharply around '60s and '70s when the abortion rights movement peaked as the linked Ngram Viewer shows.
the developing young in the uterus, specifically the unborn offspring
in the postembryonic period, which in humans is from the third month
after fertilization until birth
[Miller-Keane Encyclopedia and Dictionary of Medicine, Nursing, and Allied Health, Seventh Edition]
Unborn baby and unborn child could be synonymous with fetus. Depending on context, baby could also mean a human fetus as defined in Dictionary.Com. However, it would be very difficult to pinpoint when baby started to mean a human fetus as baby has a lot of different meanings including attractive young woman.