The following problem came up while attempting to answer the following: What does $arg(z-z_1) - arg(z-z_2) = \phi$ represents. While, OP themselves tried a different method, one could interpret the equation as relationship (region) between $2$ rays within the complex plane.
Let us start of with the simplest case, $z_1, z_2 = 0$.
Hence, the relation $\arg(z - z_1) - \arg(z - z_2) = \phi$ would look like:
The confusion occurs when $z_1, z_2 ≠ 0$. Given $z_1, z_2$ are arbritary, I am able two formulate to different scenarios as shown below
$\mathbb{\text{Fig. 1}}$
$\mathbb{\text{Fig. 2}}$
Reason for Figure 1; The relationship only restricted the angular region, which is dimensionless, but never restricted points below $z_1, z_2$, hence shaded region stays the same regardless of the ray translation.
Reason for Figure 2; The shaded region is defined by the ray arguments itself, which does not exist if the ray relationship does not exist below $z_1, z_2$.
I am unable to see which one is correct.
Here I'm Attaching two Images in which I've plotted Locus of $z$ because I don't know how to plot them Digitally. Arrow Must end at $(z-z_1)$ and must originate from $z-z_2$.
Also I am considering Principle Argument that is $(-\pi,\pi]$.