I quote from a family history book entitled The Playfair Book or Notes on the Scottish Family of Playfair authored by Arthur Grace Playfair, published in 1932 in Tunbridge Wells, Kent, England by C. Baldwin:
Elizabeth (Betsy) Bingley, elder surviving daughter of William Playfair and his wife Mary Morris, married, at a date unknown, a man described as Professor Bingley whose identity is uncertain. In the parish records of St. James Westminster the baptism of their daughter Amelia in 1818 records the father as William Bingley.
I only have a reference to this book- there are no copies here in Canada as far as I know. This Elizabeth Playfair had a second child born in 1822 named Charles and they emigrated to Canada in 1843 with her brother John Playfair and his family. She was a widow at that time and blind from birth.
I have narrowed down the William Bingleys who were buried in London from c . 1823 to 1843 and came up with 8 or 9. One of these could possibly be a "Professor":
buried March 1823 Rev. Wm. Bingley of St. George Bloomsbury, age 49, married in 1812 and had a young son born in 1813. Author of many books. Professor?
buried 1826 Drury Lane age 48. Possible brother to a John Bingley who had "Wine Vaults" there in an 1811 directory. Cannot find a marriage for him. Professor?
buried 1827 Tavistock Street Covent Garden. Linen draper, married no children. Age 73.
buried 1831 St. Saviour Southwark, Abode- Bethnal Green. age 40. No marriage record found.
buried 1838 Ossulston St., St. Pancras, age 58, married no children. Carpenter by trade.
buried 1842 Queen Street, Brompton, Kensington, age 82, widower in 1818, solicitor/attorney.
buried 1843 Queen St., Brompton, Kensington, age 47, son of entry above, solicitor. Married in 1827 and had children.
William Bingley Chip Hat Maker/Weaver death year/place unknown. 1818 accused of theft- not guilty. 1819- larceny confined 6 months and whipped. 1820- stealing a leather pocket book- one month in Newgate. 1823- convicted of manslaughter of George Durham at George Tavern, Whitechapel. One month in Newgate. (Source- Old Bailey online).
So who was the Professor? Toss the dice.