Weak incidence colouring
I don't know if this problem has another name, but it seems like it's easy to solve.
We first see that a weak incidence coloring of a graph $G$ corresponds to an edge coloring of the graph $G'$ obtained by subdividing each edge of $G$ once. This, in turn, corresponds to a vertex coloring of $L(G')$, the line graph of $G'$. This line graph consists of a collection of cliques, one for each vertex in the original graph $G$. Each vertex in each clique is adjacent to one and only one vertex in another clique.
Now, it is not hard to prove that $L(G')$ is a perfect graph by the strong perfect graph theorem (neither $L(G')$ or $\overline{L(G')}$ have any long odd cycles). This implies that the chromatic number of $L(G')$ is equal to the size of the largest clique in $L(G')$, which in turn implies that the weak incidence coloring number of $G$ is equal to $\Delta(G)$.