On wikipedia, the normal crossing divisor is defined to be (by my understanding):

(Assume $X/k$ be a smooth geometrically integral scheme of finite type over a field $k$).

Let $D = \sum_{i=1}^n C_i$ be a Weil Divisor, here $C_i$ are irreducible closed subsets of codimension 1 of $X$. Endow $C_i$ with the reduced scheme structure (hence they are integral closed $k$-scheme of $X$ of codimension 1.) We call $D$ is a normal crossing divisor if each $D_i$ is smooth over $k$ and $D_i$'s intersect transversely.

But in somewhere, I saw the notion "strict normal crossing divisor". For example,

Definition 1.5.1, p.8, in http://math.arizona.edu/~swc/aws/07/KedlayaNotes10Mar.pdf

or

5.1, p.16, in http://www.uni-due.de/~bm0032/publ/TubNbdDocumenta.pdf

The definition looks the same as "normal crossing divisor".

I would like to know what's the difference between these two definitions. It would be great with examples. Also, if the base scheme $S$ is not a field, then is the definition the same as above with the modification "each $D_i$ is smooth over $S$"?

ncandsnc, for instance in considering the singularities of pairs $(X,D)$. For more details see my answer below. $\endgroup$