Davidson are proving difficult (or rather, slow) to investigate. They were in Trafford Park, Manchester, and bodied vehicles for Ford (also in Trafford Park) and supplied the van bodies to Morris for their 8/10 cwt chassis 1924-6.
However, during my search I found
this interesting page on a Mr William Davidson and his early Daimler motor bus in Bishop Aukland, which states he moved to Erdington in 1906 - and is found in the 1911 census as a coal merchant at
139 Gravelly Lane with wife Mary Ann and daughters Kate Blanche (9) Iris (7) and Clara Gertrude (4). The latter was born Erdington, all the others County Durham (the two elder daughters at Hgh Grange, fairly conclusive proof it is the same family).
Now, W.W. Greener lived in Erdington, and ran a 10 seater Daimler on his short-lived bus service. I know 1902 and 1906 aren't the same, but allow for a little error in the dates - and is the Greener Daimler formerly the Davidson one?? By the looks it is (at a squeeze!) a 10 seater.
The writer of the article, a Mr Dukes, must be the son of Iris Davidson who married a Joseph H Dukes in Birmingham North district, 1st quarter 1930.