I found the following definition.


A Weil divisor $D = \sum\_{i}D\_i \subset X$ on a smooth variety $X$ is simple
normal crossing if for every point $p \in X$ a local equation of $D$
is $x\_1\cdot...\cdot x\_r$ for independent local parameters $x\_i$
in $O\_{p,X}$. A log resolution of the pair $(X,D)$ is
a birational morphism $f:Y\rightarrow X$ such that $Y$ is smooth and $f^{-1}D \cup Exc(f)$ is simple normal crossing. 


From this definition it seems to me that if $D = L\cup R\cup C\subset\mathbb{P}^2$ is the union of three lines passing through the same point, then $D$ is simple normal crossing. 


On the other hand I found an example showing that if $C\subset\mathbb{P}^2$ is a cusp, in order to get a log resolution we have to blow-up three times. However after two blow-ups we end up with the strict transform $\tilde{C}$ of $C$ and two exceptional divisors $E\_1, E\_2$ intersecting transversally in a point $p = \tilde{C}\cap E\_1\cap E\_2$. Why is not the divisor $\tilde{C}\cup E\_1\cup E\_2$ simple normal crossing ?



Simple normal crossing divisors
You gave the definition of normal crossing divisor. The definition of simple normal crossing is the following.


A Weil divisor $D = \sum\_{i}D\_i \subset X$ on a smooth variety $X$ of dimension $n$ is simple
normal crossing if any component $D\_i$ is smooth and for every point $p \in X$ a local equation of $D$ is $x\_1\cdot...\cdot x\_r$ for independent local parameters $x\_i$
in $O\_{p,X}$ with $r\leq n$.


What goes wrong in the case of three line passing through the same point in $\mathbb{P}^2$ is that you need three local parameters in a neighborhood of $p$ and $3 > 2$. The same for the cusp.


More generally, let $D = \sum H\_i\subset\mathbb{P}^n$ be a reduced divisor, where the $H\_i$'s are hyperplanes. Consider the points $h\_i\in\mathbb{P}^{n\*}$ dual to the the $H\_i$'s. Then $D$ is simple normal crossing if and only if the $h\_i$'s are in linear general position in $\mathbb{P}^{n\*}$. You see that three lines through the same point in $\mathbb{P}^2$ correspond to three points on the same line in $\mathbb{P}^{2\*}$. Therefore this case does not work. More generally, for a curve $C$ on a surface, simple normal crossing means any irreducible component is smooth and $C$ has at most nodes as singularities.


