Can anyone solve this riddle for me please???

In 1838, Henry William SMITH was born in Fisherton Street, Fisherton-Anger, Wiltshire to William Dutton SMITH - a Currier, and Betsy Mold. The birth was registered by Betsy.

I was a little confused with Betsy's different last name, so I headed to the census.... I could not find William Dutton SMITH on the 1841, but I did find a Betsy Mold living in Fisherton Anger with her husband THomas - a Lawyer, and their family, including one William aged 3!!!!!

I cannot find either Betsy or William Dutton SMITH in 1851, but have now found William Dutton SMITH - a Currier, living with his wife - Elizabeth of Salisbury, Wiltshire, and their daughters Louisa and Fanny in Surry in 1861. If this is the correct Henry William SMITH, he had emigrated to Australia in 1859- his shipping info states he was born in Salisbury but was from Middlesex....

SO!! Why does Betsy have a different name and family in 1841, and how does she end up with William in Surrey in 1861?? If this child Henry William was illigitimate, surely she would have registered it under her husbands name, not the real father?? How do I get to the bottom of this complicated riddle please??
HELP!!