Part 4 updated
BTW, about the lambskin not being a grimoire, to me the most covincing argument is that, I don't honestly think that people that know about it, like Fiamma or Elizarina would just call that "patchments" or "Lambskin" or something instead of
grimoire. Even the Angel didn't called it grimoire, and the Angel can't "choose" what to call it, as it can't express preferences,
so it is what it is (something that hasn't been explained yet at all and seems more important that any grimoire shown so far).