Please be more careful. You get some translations right on or at least very close, and others are just way the heck off. Danbooru policy is that no translation is better than a bad translation, since the latter spreads misinformation, but you obviously know enough to get some stuff just fine. Just double-check yourself, please.
Well, going by the mythological basis, Kasen held more authority than Yuugi... But who knows.
Updated
What does "Yama no shiten'nō" mean?The oni were the number one big shots on the mountain, by the way.Hey, you two!That's our Cirno!Really?That's crazy! That means that you two...!...there were four who were considered the strongest, and that's what they were called.Oh...Were the Kings of the Mountain!?Ah, that. A long time ago, among the oni who lived on Youkai Mountain...In other words, we big Four were like the controllers of the mountain.