Say I have an integration
$$\int_{L_1} f(z)dz $$
that I want to write as a sum of
$$\int_{L_2} f(z)dz \quad and \quad \int_{L_3} f(z)dz $$
$L_1,L_2$ are positively oriented. Suppose $L_3$ be defined as positively too:
If all contours are positively oriented do I get this:
$$\int_{L_2} f(z)dz + \int_{L_3} f(z)dz $$
or this:
$$\int_{L_2} f(z)dz - \int_{L_3} f(z)dz $$
Why?
$$\int_{L_1} f(z)dz =\int_{L_2} f(z)dz+\int_{L_3} f(z)dz $$ Because you have to keep the order of visit of the points on $L_1$.
Note that the segment you had (splitting the figure in two) is visited in both orientation and therefore is not counted.