Does Satan even understand Yuri's dream of peace? Yuri (mum of the twins) falls for Satan and they both dream of the two worlds to be 'one' where demons and humans can be at peace. She gets pregnant with twins and instead of Satan taking care of this kids, he uses their bodies as vessels to join the world together. 

I am pretty certain this is not what Yuri wanted and I feel that makes her quite dumb to trust Satan and pity him when his idea of peace is only going to be a war with demons dominating the world. Plus he obviously doesn't care about his sons and I doubt Yuri will want that if she were alive. 

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.