Maybe this is partly a confusion of terms. If $\Delta$ gives the function that governs the nonresponse (i.e. $\Delta=1$ if we have a response and $\Delta=0$ otherwise) and the variable that is subject to nonresponse is $Y$ and we have some some variables $X$ that are not suject to nonresponse (and maybe related to $Y$), then we have MCAR (missing completely at random) if $$\Delta \perp Y $$ i.e. the mechanism that drives the nonresponse is independent of the level of the response variable.
If we have MAR (missing at random), then the missingnes is not really random, so we DON'T have $\Delta \perp Y $ but
$$\Delta \perp Y|X $$
This means that given we control for the variables $X$ we have captured all relevant information about the missingness mechanism. The last form is MNAR (Missing not at random) and means that even $\Delta \perp Y|X $ does not hold, maybe because $\Delta=f(Y,X,Z)$ is a function of additional variables $Z$.
Now in reality we cannot say what is true, but MCAR is very unrealistic and in case we have MNAR we are stuck (there are some sophisticated techniques for that), so we try to include variables that are related to the nonresponse (i.e. the $X$ are related to $\Delta$). And this explaines why Factors that are known to have influenced the occurrence (or non-occurence, $\Delta$) of the response (stratification, reasons for nonresponse) are to be included.
