I cannot help with the Baum/Baun confusion but it is my humble opinion that it was an emigration miscue. However, I can confirm from documents passed down by my mother (Laurena Etta Baum Rychener) that Grampa George Baum was Anna Erbskorn's son along with sibs Alfred, Etta J, Frieda, Carrie, Ervin and Elmer. These documents include the Switzer-Dietz Family files, dated Aug 1999. It is also my opinion that Anna's full name was Anna Gertrude Erbskorn and that her mother was Barbara Dietz Erbskorn who was married to John (Johannes) Erbskorn.
Thank you for confirming that Grampa George's father was Gotlieb. I had the name penciled in but with so many Anna's and John's in the family who can be sure - I had to check the birth dates to realize that there indeed was a generation gap.