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: the components of $D$ should also be smooth. What you have is justnormal crossing. $\endgroup$