Daisy's relationships with the two men are both shallow. With Winterbourne, she has a connection of being Americans in a foreign locale. However, he is also more of a protector and does not want to see her being taken advantage of. He has placed himself in a position of authority over her by nature of their common origin.
With Giovanelli, however, she treats him more of a combination tourist spot and means to provoke Winterbourne. She and the Italian do not take their relationship seriously and don't intend to. With Witnerboure, while Daisy does not take it seriously, Winterbourne does. This puts Daisy into a position of trying to maintain her independence against Winterbourne's overprotectiveness.