This recipe is pretty close to what I grew up with, where you cook the dutch baby
in the butter rather than mixing the butter into the batter. Since the first thing we did on removing from the oven was sprinkle liberally with sugar it was important that it be "wet" so that the sugar would dissolve and stick.
No idea on the "baby" part of the name. I remember I tried to find that out a couple years ago and couldn't (it is hard to filter out the Dutch baby name sites that keep coming up).