"Weak" signifies an indeterminate notion; there are several different specific versions of weak category. It can also be used as a vague notion, when one is merely looking at what one would like to have. There are specific kinds of category, such as "Batanin category" [B1], "multitopic category" [HMP1,2,3,4,5], [M8]. When one wants to talk about the "ordinary, strict" version of the notion, one says "strict". Thus, "strict category" is my term for an ordinary, strict ω-category.