This is just a story that Persephone makes up. She likes to play with the thought that she in an earlier existence was a powerful Underworld goddess, though no one is known to have ruled the Underworld before Hades. This also is a little reference to the theories suggesting that Persephone once was the powerful and independent ruler of the Underworld and a much older deity than both Hades and Demeter.
The etymology for Persephone is very uncertain. The existence of so many different forms of the name have led some scholars to believe that the name has a pre-Greek origin and that the Greeks had some difficulties to pronounce it. That it should mean "Destroyer of Light" or "Bringer of Death" is only a suggestion among others. An alternative interpretation is that the first part of her name should be "p(h)erso" rather than "p(h)erse", and that her name means "female thresher of grain".