The difference between our quantum system and the energies of the toy model is that in the quantum case, only three energies are allowed, meanwhile in the classic case of the toy model, all range of energies from the highest to the lowest are permitted. This happens because we are considering that any angle between the nuclear spin and the electric cloud is possible, but in quantum mechanics only tetta=0, 90 or 180 is allowed (m=1,0,-1).
In order to move from a quantum system to a more classic system, we would need some restrictions on the values of the Hamiltonian. The spectroscopic quadrupole moment Q should be very small, indicating that there is almost no deviation from spherical shape in the nucleus, and the same for the Vzz component, which would lead to a weak field gradient. With this two conditions, our system is less dependent in the direction because have a lot of symmetry, and the splitting due to the quadrupole operator will be too small to be measured. Then, the different energy levels will be seen for us as a continuous range of energies, as in a classical system.