There is only one coherent state: $$|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle $$
Also, a pure state does not mean a coherent state.
But what does one mean when they talk about a coherent superposition of ground and excited state:
$$c\left|g\right> + d \left|e\right>.$$ Drawing on the bloch sphere, it is on the surface, but so does just being a superposition. But what does one mean and what does it imply?
I also see a Phys.SE post: 262052
The word "coherent" is used in Physics in a rather sloppy way. Your first state is a linear combination of harmonic oscillator eigenvectors that turns into a gaussian in momentum/position representations. In a more general background, a coherent state is just a state where coherences (off-diagonal terms in the density matrix) are non-zero, which means the state can skipp from one stationary state to another.
Now, a coherent superposition is quite like a coherent state: a superposition is said to be coherent if there's an observable that, if applied to one state, can turn it into another also present in the superposition. As an example, consider the $z$-axis spin up and spin down states of the electron in a Stern-Gerlach experiment. Then there is one spin operator, namely $S_x$, that can turn one into the other. This means they form a coherent superposition. As a counter-example consider the ground and the first excited states of the harmonic oscillator: the creation operator can turn the former into the latter, but this operator is not an observable. The superposition is a non-coherent one, meaning that off-diagonal elements in the density matrix are irrelevant to the problem at hand.
Coherence has many faces. See Quantum coherence - what is it's definition?
First state refers to a state of the field. Originally Glauber developed this formalism to give quantum description of laser fields. Later it was adopted in other fields.
Second state refers to the state of a 2-level system (in your case). You can also get superposition states with incoherent light, but those are not very useful. The word coherent is used to describe the superposition states that you create with coherent fields. Usually you don't have to quantize the field, but can still work in what is know as "semi-classical" approximation. This means that you have aclassical field and a quantized system. This is the more common experimental situation that one encounters. The field dealing with this is called "coherent control". Check out P.L Knight or N.V. Vitanov, they have planty of papers there.
