The members of AC are called anodyne cofibrations and the members of AF are called anodyne fibrations (as in anodyne morphism).

Anodyne and trivial (co)fibrations

The notion of premodel category doesn’t come with a good general notion of weak equivalence. But if a particular premodel category has a good notion of weak equivalence, such as one of Barton‘s relaxed premodel categories, one needs to distinguish between two types of cofibrations (and analogously between two types of fibrations):

An anodyne cofibration is a member of the class AC

A trivial cofibration is a cofibration that is also a weak equivalence

In principle one must also distinguish a third class of cofibrations that have the left lifting property with respect to fibrations between fibrant objects. However, in a relaxed premodel category, these are trivial cofibrations. (Barton, Prop 3.5.2)