I think a lot of the information in the referenced profile is conjecture. A Baumgarten is an orchard and a Baumgärtner or Baumgartner is somebody tending to it. Bumgarner would at best be the name written as heard from somebody speaking heavy dialect. And the "d" in Baumgardner seems to me to be an Anglicism. Absent any information about the birthplace in Germany the best chance of getting close to a correct spelling seems to be to start with the presumptive Swiss origin: The Bumgarner profile mentions Langnau in Switzerland.
There are several municipalities and communities have the name Langnau in Switzerland:
- Langnau am Albis, in the canton of Zürich
- Langnau bei Reiden, in the canton of Lucerne
- Langnau im Emmental, in the canton of Bern
- Langnau bei Märwil, in Affeltrangen municipality, canton of Thurgau
Swiss citizenship is derived from citizenship in a home town. The home town keeps the family registry, no matter where in Switzerland a person lives. The Familiennamenbuch der Schweiz contains family names of Swiss citizens, their home towns, the time of first appearance of the name, and, if known, where they came from, provided there are still Swiss descendants living in 1962. A first step could be checking the Familiennamenbuch for Baumgartner and spelling variants to see if any have Langnau as home town and which of the different Langnaus it would be. If there are no living descendants one would have to try to search the records of each of the Langnaus.