Based on that page, Chizuru is written in kanji (??) when she's Mizuhara, and in hiragana (???) when she's Ichinose. The reading (the hiragana characters in the parentheses) is still ??? (Chizuru) in both cases though.
What I've been trying to say is that the distinction between Chiduru/Chidzuru/Chizuru in this case is an arbitrary translation choice like with Osama/Usama bin Laden. Either is technically correct because romanisation is ultimately just a rough approximation.