I just realized, Satan does take action (sending Shiratori) earlier than either Lucifer or Mephisto finished preparing his body. However, Shiratori (I mean the demon possessed him) was unsure that Rin is the target, which means Mephisto didn't tell Satan the exact position of his demon son. Likewise, Lucifer had to examine Yukio's eye by himself, which means Satan didn't tell Lucifer this connection. Therefore, all three of them don't trust each other. Then we have at least three major players each have their own agenda in the whole story ...