How to write singlets and triplets in second quantization for fermions?

It has been a long time I haven't done this and I am having a hard time writing things down in second quantization notation. Let us have a $$n$$-body system where the spin part and orbital parts are decoupled.

If want to write a two particle state, let's say a triplet state $$\alpha$$, with parallel spins, an electron with energy $$\epsilon_1$$ and another with $$\epsilon_2$$, I would write something like $$|\alpha\rangle=c^\dagger_{1\uparrow}c^\dagger_{2\uparrow}|\emptyset\rangle$$ where $$|\emptyset\rangle$$ is the void state, and $$\uparrow,\downarrow$$ are the spin states.

Same for a singlet $$\beta$$ with two electrons in the same energy $$|\beta\rangle=c^\dagger_{1\uparrow}c^\dagger_{1\downarrow}|\emptyset\rangle$$

But how do I distinguish a state $$\gamma$$ with $$m_z=0$$ (different energies, opposite spins)? What does this do? $$|\gamma\rangle=c^\dagger_{1\uparrow}c^\dagger_{2\downarrow}|\emptyset\rangle$$ This could be either a triplet (with opposite spins) or a singlet, what am I missing?

Edit: I now realize that there has to be a difference in the Fock space, between writing $$|n_{1\uparrow},n_{1\downarrow},n_{2\uparrow},n_{2\downarrow}\rangle=|1001\rangle$$ and $$|0,1,1,0\rangle$$ but I don't know how to interpret these states in terms of the singlet and the unparalleled triplet

The point is that the wavefunction is symmetric in spin for the triplet and antisymmetric for the singlet. So in your notation, the triplet is $$\frac1{\sqrt2}(c^\dagger_{1\uparrow}c^\dagger_{2\downarrow} + c^\dagger_{1\downarrow}c^\dagger_{2\uparrow})|\emptyset\rangle$$ and the singlet is $$\frac1{\sqrt2}(c^\dagger_{1\uparrow}c^\dagger_{2\downarrow} - c^\dagger_{1\downarrow}c^\dagger_{2\uparrow})|\emptyset\rangle.$$