Further research has revealed that Samuel Frederick Baumgarten actually went by Samuel Henry Baumgarten for the rest of his life... In my research into the Baumgarten family I had come across Samuel Henry Baumgarten and his descendants regularly, but had found nothing to link him to my Baumgarten family. Actually, the only way I was ever able to work out that he was the same as the person who was baptised Samuel Frederick Baumgarten was through reading some centuries-old legal documents at the National Archives UK.
It would seem that one of the reasons why I didn't realise Samuel Henry Baumgarten was related to my Baumgartens - even though Baumgarten was an extremely uncommon surname in 18th and 19th century London - was that he was disowned by the rest of his family over a dispute over an inheritance.
Samuel Henry Baumgarten was the firstborn of Samuel Christian Frederick Baumgarten and Mary Joynes. His maternal grandfather, Henry Joynes, died in 1754, leaving an inheritance to his two surviving children: his son Samuel Joynes and his daughter Mary, Samuel Henry Baumgarten's mother (or her children in equal shares in the event of her death).
Samuel Joynes died in June 1770, leaving the residue of his estate to his sister Mary, and in the event of her death, in equal shares to her children. However, Mary died less than a month after her brother, and being intestate, and following the law of primogeniture, her firstborn Samuel Henry Baumgarten inherited everything.
Taking into account the wishes of Henry Joynes and Samuel Joynes, the rest of the Baumgarten family, lead by their father Samuel Christian Frederick Baumgarten, obviously felt they had a case for contesting the allocation of the entire inheritance to Samuel Henry Baumgarten, and they took him to court. A succession of court cases over many years followed, which eventually resulted in the inheritance from Mary Baumgarten née Joynes being split equally amongst her surviving children. Sadly, it appears that relations between Samuel Henry and the rest of his family irreparably broke down as a result of these court cases.
You wouldn't necessarily expect to learn too much relevant to family history from the sometimes huge pieces of vellum that the court cases were written up on - or at least I naively didn't! However, as the court cases took place over many years, it was possible to track the births and deaths of children and grandchildren of the Baumgarten family over that time. I could work out which of the known children (from baptism records) died young, because they were or were not included in the parties bringing the case to court. I could also confirm Mary Baumgarten's death date, which I had not been able to find with certainty anywhere else. So its worth poring over ancient documents which might seem virtually impenetrable because of the old legalese and handwriting because you just might find some very useful facts!