In this video he says "put baby in" and not "put the baby in" Also, in other baby type videos they say "place baby inside" or "now you can see that baby and mom are happy"
There is no "the baby" or "the baby's mom" or "the mom".
The video is one example of many. I asked someone who said it is something to do with the agenda of keeping people happy and not offended by mentioning a gender. I don't think this is true though.