This is not a situation where different surnames were used, but rather a case of no accepted standard for spelling this name. If I were writing an article about this family, I would use a standard spelling, but would also provide transcriptions of the church records with the exactly spelling of all text used in those records.
Wikipedia has an interesting article on German orthography. An interesting topic is how names appear on German passports. In some areas of the passport, letters in names such as ä, ö, ü are used, and in other areas of the same passport, they become ae, oe, and ue. The issue in this case is non-machine readable (with the diacritical marks) or machine readable (without).