Anyway, since F is moving, again, some information that I know I told some of the players, but not all.
The Church is essentially three branches, the Priesthood, the Crusaders, and the Inqusition.
The Priesthood is what it sounds like, and made up of priests, bishops, and so on. The head of the priesthood is the Vicar of St. Cuthbert and the head of the entire church.
The Crusaders are the military arm of the church and professional soldiers and whatnot. Headed by the Crusader General.
The Inquisition is what it sounds like. The secret police of the church who control the populace through fear and torture, and purge heresy and apostasy with fire and sword. Witch Hunters are the common name for younger, lower ranked, Inquisitors who are tasked with monitoring and policing the laity and with handling matters of lay heresy, witches (Witches, Sorcerers, and unsanctioned Psions and Wizards) and with killing abominations (Tieflings, Sorcerers, and other such things). Heresy and apostasy within the church is handled by the older and more experienced and high ranking Inquisitors. Laure is technically a Witch Hunter, hence why she's leading F against a cult made up of lay people and not the church.
The common metaphor is that the Priesthood is the head or heart of the church and guides it, the Crusaders are the sword and bring the truth and justice of the Saint upon heathens and His enemies, and the Inquisition is the shield of the church, safeguarding it from heresy and the subtle taint of the Enemy.
The divide between the priesthood and the other two branches of the church is blurry. Priests like Vivien often get attached to help out the Crusaders, and the Inquisition generally recruits from it. So people in service to the church don't really see themselves as belonging to any one group - besides the Inquisition, who are in it for life and fairly insular for obvious reasons.
Regarding Laure, the idea is that she's conservative even by Cuthbertine standards. Thinks all arcane magic and alchemy are blasphemous and ought be banned. It's a popular strain of thought among some of the Inquisition and the church in general. Arcane magic was always seen as a bit blasphemous, if necessary, and alchemy/transmutation in particular as bad - the vanity to feel that mortals can and should remodel reality. The world is as the gods have made it and so it should be. Plus from a more cynical standpoint, the church likes having a Clerical monopoly on magic, and the obvious fear that scholarly Wizards might figure out the truth that reality is malleable, the gods dependent on mortals, and all that nasty truth stuff.
Good Cuthbertines know better than to let truth get in the way of the truth. Our eyes may deceive us, but our god never will.