It's not really that complicated. The mask is the demon. In Cio's case, her mask was destroyed and she was killed, but Praman Nand rebuilt her mask and most likely gave her new names, or got someone to give her names, or whatever, it's not that important. The point is it's like she suffered a serious brain injury. She remembers being her old self, but not everything about it, and in a sense, she isn't her old self anymore. Same mask, same memories, different self.
Princess, however, is a different entity than Vladok, since his mask broke and he was unleashed as a rampaging black flame, but Allison gave him a new mask and a new set of names. Because it's the same black flame that was masked, Princess remembers what she used to be, but most likely not all her memories, and is now a completely different demon all together
As for the angels, that shits even simpler. If you're a weak shit angel like White Chain, having your body destroyed could kill you outright, but Juggs is a whole other level compared to White Chain, and only suffered a single, clean cut, it destroyed his body but no way did it kill him, and Delicious is just outright shown, his body was probably destroyed or he was banished, captured and then killed, which is why he began to reincarnate on panel.
The confusion comes partly from people trying to over explain very whimsy and open ended points, rather than just accepting their vague nature. And also missing clear points.