My overarching question is why in the Artin presentation of the (geometrically defined) braid group there are no more than two relations. I learned this can be proves using the fact that the group $Br_m$ is the $\pi_1$ of configuration space $Conf_n=(\mathbb{C}^n\backslash \{z_i=z_j, i\neq j\})/S_n$ together with the fact that when computing the $\pi_1$ of a CW complex you can restrict your attention to its 2-skeleton.

Now I'm wondering if there is an explicit description of a cell decomposition for the configuration space (in particular of the 1- and 2-cells in it).