We study a specific line arrangement obtained from a generic $2$-section of the braid arrangement, and compute the fundamental group of its complement via braid monodromy. We show that the resulting presentation of the fundamental group coincides, under the identification of generators, with the modified Artin presentation introduced by Margalit and McCammond. Moreover, we extend the construction to the Manin--Schechtman arrangements $MS(n, k)$, which are higher analogues of the braid arrangement. Focusing on the case $k = 2$, we obtain an explicit presentation of $π_1(\mathbb{C}^n \setminus MS(n, 2))$.