The quantum-mechanical spin state of a spin-1/2 particle is quite a complicated beast if you want to understand it in detail.
As a first approach, it's true that for any spin-1/2 system, given a quantization axis (say, the $z$ axis), there are only two fully-distinguishable states, $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ ("up" and "down"). However, it is also perfectly possible for the system to be in a superposition state of these two, with arbitrary complex-valued amplitudes, so the most general state has the form
$$
|\psi\rangle = \cos(\theta/2)|{\uparrow}\rangle + e^{i\phi}\sin(\theta/2)|{\downarrow}\rangle
\tag 1
$$
for two parameters $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$. (Why can it be written in this form? Because it must be normalized to unit norm, and because we don't care about global phases.)
Now, here's the thing: for any state of a spin-1/2 system, of the form $(1)$, there exists an axis of quantization for which said state is the "up" state along that axis of quantization; in other words, there always exists a unit direction vector $\hat{\mathbf n}$ such that $|\psi\rangle$ is an eigenstate of the spin component operator $\hat{S}_\hat{\mathbf n} = \hat{\mathbf n}\cdot \hat{\mathbf S}$ in the same way that $|{\uparrow}\rangle$ is an eigenstate of $\hat{S}_z$. Moreover, this unit direction vector $\hat{\mathbf n}$ has polar spherical coordinates $(\theta,\phi)$.
This means that we can think of the set of all spin states of a spin-1/2 system as a sphere, which we call the Bloch sphere. This is a crucial, core concept, and it is well worth reading up on it, both at its Wikipedia page as well as in textbooks. (If you want a specific suggestion, try Nielsen and Chuang.)
So, how does this interface with NMR?
- If all you do is add a magnetic field $\mathbf B$, then it is indeed correct that the spin will precess around $\mathbf B$, in the sense that the unit-vector Bloch-sphere representation $\hat{\mathbf n}$ of $|\psi\rangle$ will precess around the magnetic-field vector.
On the other hand, if one performs a projective measurement of the spin component along any chosen quantization axis (be it the $z$ axis or something else), then the measurement outcomes will only ever be "up" or "down" along that chosen axis.
So, what constitutes a measurement? well, for starters, it requires an actual interaction with an external system (starting with fluorescent emission). But what really constitutes a measurement, you ask? That's one of the biggest open problems in the foundations of quantum mechanics.
It's important to note here that the spin precession sounds like it's a very likely candidate to the physical mechanism behind the emission, in the same way that a hydrogen atom placed in an electronic superposition state has a charge oscillation that matches the emission frequency, but that's basically a red herring. The EM field is a quantum entity, and the mechanism for the emission is quantized transitions in interaction with the quantized EM field.
Hopefully this is helpful and enough to get you started.
