Every time Rika died, she didn't just go to the next world. She went to the Sea of Fragments in-between, and created a copy of her memories to insert into a Rika in another world. Rika eventually decided in Higurashi Rei's Dice Killing Chapter (kind of an epilogue chapter) that because she is, mentally, effectively completely different from the little girl Rika, she would take the name Frederica Bernkastel (Furuderika and then Bernkastel after her favorite wine brand). However, due to the lessons she learned in that chapter, she decides to stop hopping between worlds when she dies and have this be her final life, like a regular human would. Thus, she effectively splits off from the 'her' that is still in the Sea of Fragments, becoming Rika again for real. However, the one in between still exists, and she is the one that ends up adopting the moniker of Bernkastel.
So Bernkastel is Rika, but Rika is not Bernkastel, basically.