Does Satan even understand Yuri's dream of peace?
No. He does not understand her dream at all. But you have to see things from his point of view.

He is a demon that cannot live in Assiah because nothing can host him for too long. So differently from Mephisto, he cannot understand / study Assiah at all.

From his poing of view, being in Gehenna for all eternity, the concept of peace as we humans understand it is just too alien.

We can infer that he was in love with Yuri, but he took what she told him in the few months / years they co-existed and applied his own interpretation.