Well 1916 it was the WWI. Those territories have seen some fighting between Russia and Austria - Hungary.
The birth certificate if survived would be stated according to religion. If a person in question was a Catholic you should look into records of the Catholic church from those territories', if not you should look accordingly.
By the way a city of Lwow.
would there have been any advantage of saying you were born there when possibly not?
Not in Poland. You're the only person who can answer that.