You equate "full polarization" to a known spin (state), "weak polarization" with a superposition of spin states, and "unpolarized" with "uniform (isotropic) properties". "Properties"? What happened to only limiting discussion to spin? Any (quantum) state can be equivalently described as a mixture of states. A good way to look at it, is as a vector. Any vector has components which depend on the basis vectors. That is, if one coordinate system describes a vector as (1,0), then you can choose to create a different coordinate frame of reference/system/basis by a simple rotation about the origin. In the new coordinates, the same identical vector will have x and y components different from 100% x and 0% y. (Say a 45° rotation of the coordinates, NOT of the vector, will result in the vector being (√2,√2). This doesn't change the vector at all.)
Also, apparently you are assuming your quantum particle can have ONLY two 'pure' polarizations - but I'm not sure about this. Usually, we attribute non-zero spins to each particle but this depends on the context. I'm all for +1 and -1 spins. The math is equivalent if you choose +1 and 0, instead. BUT if you allow +1,0 and -1 as possible pure states, then that completely changes the problem (and the system). Obviously, the simplest system is two state (well, the simplest is 1 state, but with one state there is nothing interesting that can happen, is there?). Spin angular momentum is quantized for real particles. I encourage you to read the Wikipedia article on Spin (quantum spin). There it discusses spins as having half integer values, so my convention above is different from its. It also states that particles spin is inherent. This is confusing if you are considering spin as being dependent on (say) field strength. You need to keep several factors in mind. One is that usually its the MAGNITUDE of the spin, not its direction, that they're talking about. The other is the fact that a "particle" can be elementary or composite. A composite particle can be viewed as a system of particles each with their own spin. In the simplest of (non-trivial) quantum systems all you can do is to flip the spin from +1 to -1 or vice versa (or equivalently from +½ to -½). There's NOTHING in between. BUT, we can think about the probability of the particle being in the +1 state - that clearly has to be between 100% and 0%. If we know nothing about the particle's spin, then a two-state particle (oops! we know that about it at least) has 50% chance of being in either state. Quantum Mechanics deals with what can be observed about a system. The math is expressed so that it provides the context for those observations. I don't think "full" or "weak" are very useful terms when discussing this type of system. For instance, let's say we have an electron (two spin states) in a device that allows us to measure spin in ANY direction. So, if we were to measure the spin along one axis and find it is +1 and then measure it along another axis, the classically trained physicist would assume the spin (if it hadn't changed - which is a whole 'nother can of worms) would be zero along that new axis. But wait! QM requires that spin be either +1 or -1. It doesn't ALLOW a spin of 0. Guess what? You won't measure a spin of 0, either! This is NOT intuitive! It is the strangeness of QM at work. (incidentially, it isn't actually possible to measure the spin and not mess with a single electron, so a more realistic experiment would be to have a device generating a stream of electrons all with the same state and using that stream in your measuring device. The logic is the same, you would never be able to find a direction where the spin was EVER anything but ±1.)
Note also that "spin" can mean different things - angular momentum, magnetic spin, orbital spin, you need to be very careful about using a single term without being clear about context. Which gets us back to your terminology again. Full circle.
