How to simulate phase separation in a discrete system using LAMMPS?
Firstly, a couple of comments.

The Cahn-Hilliard model, for which you gave a snapshot, represents the two fluid phases on a much larger length scale than the atomic scale. It's a continuum model. So, whatever particle-based model you use, you would need an enormous system to get close to similar-looking pictures. The interfaces will always look more irregular, in a practical particle-based simulation. Don't expect to see nice circular droplets.

Secondly, even that Cahn-Hilliard simulation is not yet at equilibrium. The droplets are in the process of coalescing, and the final state will be a single large droplet of one phase in the other (or possibly the two phases will form slabs, with planar interfaces between them, it depends somewhat on the boundary conditions and dimensions of the surrounding box). The driving force is the reduction of the total interfacial area. This whole process takes time. Something similar should happen in the particle-based simulations, but because the length scales are so different, the time scales will also be different.

Given all that, your first choice of models may be suitable. However, you need to choose a temperature which is high enough for both the phases to be liquid. You haven't done this! The triple point temperature of 2D Lennard-Jones is $k_BT/\epsilon\approx 0.4$. At your temperature of $0.1$, the Lennard-Jones atoms will be trying to crystallize; in practice they will get "stuck" in irregular semi-crystalline semi-amorphous clumps. Nonetheless, if you raise the temperature above $0.4$, you might see liquid-liquid coexistence for this system. I can't be sure, and you may need to experiment.

I have some suggestions for a simpler approach, if all you want to do is see liquid-liquid phase separation using LAMMPS. Firstly, it is OK for both the fluids to be identical. For example, calling the two species A and B, the A-A and B-B interactions can both be Lennard-Jones. If you choose the A-B interaction to be different, and unfavourable compared to Lennard-Jones (for example, the completely repulsive Weeks-Chandler-Andersen version of Lennard-Jones), you should see phase separation.
The advantage of this approach is that the Lennard-Jones phase diagram is well known, making it fairly easy to choose a density and temperature at which both the phases will be liquid.

Even simpler, you can choose A and B to interact through a coarse-grained potential such as the dissipative particle dynamics DPD potential
$$
v(r) = \begin{cases} \frac{1}{2}a(1-r)^2
&amp; r&lt;1 \\ 0 &amp; r&gt;1
\end{cases}
$$
where the cutoff distance is taken as the unit of length.
LAMMPS has a DPD option, and you can find a perspective article on DPD
by Español and Warren J Chem Phys, 146, 150901 (2017)
also available as a preprint. There are also tutorials on DPD readily available on the internet. 
DPD has the advantage of being very fast;
also it is sort-of intermediate between an atomic model and a continuum one,
since the particles in DPD really represent regions of fluid, 
not individual molecules.

Once again, the A-A and B-B interactions can be made identical, 
for example $a=25$, with a typical temperature $k_BT=1$
and a density $\rho=3$ in 3D; you would need to experiment for suitable parameters in 2D, but the precise values may not be crucial. The A-B interaction could be the same, but with a larger value of $a$, for instance $a=50$. 
This approach has already been used to model fluid-fluid separation,
for instance in papers by Novik and Coveney 
Phys Rev E, 54, 5134 (1996)
also available as a preprint,
and Phys Rev E, 61, 435 (2000)
also available as preprint.
Those papers are a bit involved, 
and there are many complicated technical issues with 
the DPD method.
It is not essential to go into those in depth.
If you are not essentially interested in the hydrodynamics,
it is quite satisfactory to use the DPD potential
in a standard molecular dynamics simulation with a thermostat of your choice.