Relationship between strict exogeneity and exogeneity in Time Series ADL models

I am currently learning about ADL models in Time series regression. The textbook notes down two types of exogeneity: Strict exogeneity and exogeneity. Exogeneity is defined as $$E(u_t|X_t,X_{t-1},...)=0$$ Strict exogeneity is defined as $$E(u_t|...,X_{t+2},X_{t+1},X_t,X_{t-1},X_{t-2},...)=0$$ Further, it is stated that strict exogeneity implies exogeneity. I don't understand this statement and would actually argue for the reverse: exogeneity implies strict exogeneity, or at best, that the two do not have a direct causal relationship. My reasoning for this is as follows. If we assume the very simple case of $$E[X|Y,Z]=0$$ then it is by no means clear that also $$E[X|Z]=0$$ since for this we need the assumption of Conditional Independence $$X \parallel Y | Z$$ (please note that the parallel sign is supposed to be independence, but I don't know how to get the symbol into here).

In particular, this relationship which we need to assume appears to me exactly the relationship that strict exogeneity supposedly entails. Can anybody correct my reasoning?