I discovered on this site that Diane's real name was Dianne de Poytiers. I wonder why it was transformed into Diane, with one n and the Y of Poytiers was swapped to an I.

You're absolutely right, that was the way Diane signed her name. "Dianne" is the pure French version of the name, and her family originated in Poitiers. In Old French the letter "i" was very often replaced by a "y"