Personally, my suspicion is that speculation on trying to tie Arkentools to various schools of magic is going to be largely irrelevant.
Arkentools are Tools of the Titans. Tools of the Divine. They shouldn't be bound by the rules of magic that limit the mere mortals that dabble in the arcane. These aren't magic objects, they are Artifacts. They don't break rules, they are above the rules.
“It izz written!” bellowed Beelzebub.
“But it might be written differently somewhere else,” said Crowley. “Where you can’t read it.”
“In bigger letters,” said Aziraphale.
“Underlined,” Crowley added.
“Twice,” suggested Aziraphale.