>>both for sweet (tarte) and for savoury (quiche, pizza) concoctions
Aha, thank you. No wonder, then. (Why did I not remember that from any of the endless previous discussions on pies and cakes?)
So let's see if we're closing in on it:
AE: pizza/pie/quiche crust (bottom, edges, & top where applicable)
BE: pizza/tarte/quiche base (mainly bottom) / crust (mainly edges)?
If we need to distinguish between the different parts of the pie crust, we just say bottom crust, top crust (in a two-crust pie only), and edges, or perhaps outer crust. There should be plenty of examples in internet recipes for pie crust.
As for 'cook,' yes, we do use it as the most general word for what is done to any raw ingredients. We only say bake, boil, fry, roast, grill, sauté, steam, poach, etc. when the specific method of cooking matters, as in a recipe.