Fred, your situation is precisely the same as my own and that of the children and grandchildren of many immigrants. I know for a *fact* that my grandparents were born in Lithuania and I know *precisely* what their surnames were. But different records say different things. Some documents say they're from Russia, some say Lithuania. Sometimes their surnames are shortened, sometimes spelled different ways. Even first names vary.
In addition, until somewhat recently many people used their ethnicity rather the place of their birth when asked their nationality. So a person who was an ethnic Lithuania who was born outside of the formal borders of Lithuania at that time might still say they were Lithuanian.
When you are working with records prior to 1900, this is just how it is.
Borders were fluid. Spelling was fluid. Many immigrants weren't literate. People often Anglicized their names in order to fit in.
In the case of my family, we used an educated guess as to the village in Lithuania they came from and we wrote to the state archives and paid for a search, using the correct Lithuanian surnames. We got some amazing records that took us back a couple of generations.