How is spin-state of an electron defined? An electron possesses ‘spin’ even at absolute zero. But how are the spin-states identified? It couldn’t be in terms of ‘energy’ states because well, the electron does not possess any energy at all. 
 A: The explanation of the electrons spin in the 20th of the last century was the next:

: ... the magnetic moment of the electron has a straightforward explanation. The rotation of a body with a charge on its surface necessarily entails a circulation of electric charge, or in other words an electric current. The circulating current gives rise to a magnetic field in the electron just as it does in the windings of an electromagnet. Both the spin angular momentum and the magnetic moment can be represented by vectors, or arrows, lined up along the spin axis. By convention (and because the charge of the electron is negative) the vectors representing the spin and the magnetic moment point in opposite directions.

The isolated electron by Philip Ekstrom and David Wineland

As I know never the story was told in the other direction: The electron obey a magnetic dipole moment and the deflection in external magnetic field has to be associated with a spin in analogy to the gyroscopic effect.

How is spin-state of an electron defined?

The spin is a constant, without a +/- sign. If you talk about a spin-up and spin-down, you have to be aware that you are talking about the orientation of the magnetic dipole moment and the associated spin in atoms. In the s-shell the two electrons (of the Helium) are oriented in opposite directions. “In opposite directions” is the better description as to say spin-up and spin-down. Because if yo turn head-over, the two electrons switch their up and down.
Now I have two answers for you:


*

*An electron near absolute zero temperature has very low kinetic energy, it is nearly motionless. In this case the magnetic dipole moment still exist, it is an intrinsic property of the electron, but the observation of deflection / spin in an external magnetic field is impossible. So somehow you are right for free electrons.

*But the temperature near zero is very important for Bose-Einstein-condensates. The condensate is not longer under the influence of the chaotic movement of the atoms due to their temperature and this gives the atoms the possibility to align their magnetic dipole moments to a lattice-like structure and to act as an unit.
A: An electron does not have a spin at absolute zero, because no system can reach absolute zero. Why you even brought this into your question, is unclear as well because it has nothing to do with defining an electron spin. The inability to get a system to absolute zero is known as the third law and if you were able to do it, you would find that there would be no motion.
