Walters and Hanna's daughter is mentioned three different names in the course of the story: Sabeth, Elsbeth and Elisabeth. Who gave her these names, and explain why the different people call her that?

From the text:

Walter: Why did I call her Sabeth? A question in response to my question, Why Elsbeth? In between came her statement, That’s the theatre of Dionysos. Why did I call her Sabeth ? Because Elisabeth seemed to me an impossible name. In between, remarks about broken pillars. Why Elisabeth of all names? I should never call a child Elisabeth. In between, traffic lights and the usual hold-ups. Well, that was her name, her father had chosen it, there was nothing to be done about it now.

Sabeth's mother calls her Elsbeth.