I can only vouch for usage here in the Bos-Wash Corridor...
"The South" by itself, in the US, is usually taken to mean the states which made up "the Old South", from Virginia going southward, remaining East of the Mississippi River. (Also sometimes referred to as "the Deep South", at least by us Northerners -- which, again, is often taken to mean North-Eastern unless otherwise stated.)
Thus, "A Southern (US) accent", unless otherwise qualified explicitly or by context, does not imply all Southern states. Texas would be more correctly described as Southwestern... at least, in the dialect I speak.
(It has been observed that which Northern Florida is culturally attached to Georgia, Southern Florida is more closely associated with the Northeast ... and thus, within Florida, one goes North in order to Go South and vice versa.)