"un sofa" is slightly different from un canapé in terms of structure. "Un sofa" is like a bench with a back like that of a chair (there is often a void between the bottom of the back and the seat), and it can have arms or not.
"un canapé" has a full back and two full arms. It matches what your mother calls a davenport.
"un Chesterfield" is covered in leather and has specific features (criss-cross relief).