The key point is the algebra of your observables and the resulting group.
If you work with $\hat x,\hat p$ and $\hat 1$ this is the HW group, and Perelomov shown that translating the h.o. vacuum state gives the same as the original Glauber coherent states. Indeed in this case the CS is just a Gaussian displaced so that $\langle x\rangle$ and $\langle p\rangle$ need not be $0$.
If you work with angular momentum operators then the translation is a translation on the sphere, i.e. a rotation, and CS are rotated “ground states”. The ground state is either the state with $m=j$ or $m=-j$.
This is the essence of the Perelomov coherent states: you have a fiducial state and you act on it by a generalized displacement, which is a group transformation, and the group comes from the algebra of observables. Thus one can define $SU(1,1)$ coherent states, as discussed in Perelomov’s book, and pretty much any type of coherent state in this manner.
There is another key feature. It is known that, if the fiducial state is highest (or lowest) weight for the irrep of the group, then the whole geometrical setuo comes with a natural Poisson bracket on the appropriate manifold, i.e. the CS also naturally live in the classical phase space for this system since they are parametrized by points in the phase space. (The manifold is actually a coset space closely tied to the invariance properties of the fiducial state.). Hence these nice drawings either in the plane (for HW) or on the sphere (for angular momentum) where CS are mapped to localized lumps centered at the phase space coordinate of the CS.
There’s a nice review
Zhang, Wei-Min, and Robert Gilmore. "Coherent states: theory and some applications." Reviews of Modern Physics 62.4 (1990): 867.
and Bob Gilmore has published quite a bit of pedagogical work in the 80’s, v.g.
Gilmore, Robert. "Coherent states for bosons and fermions: A tutorial." Progress in Particle and Nuclear Physics 9 (1983): 479-494.
There are quite nice geometrical generalizations, such as vector coherent states, but these are much more sophisticated: see
Bartlett, Stephen D., David J. Rowe, and Joe Repka. "Vector coherent state representations, induced representations and geometric quantization: II. Vector coherent state representations." Journal of Physics A: Mathematical and General 35.27 (2002): 5625.
Saturation of the Heisenberg relations is a bonus and is “baked in”: fiducial vectors have this property in general and you’re just displacing this vector so with some suitable definition of observables one easily recovers this property. See
Delbourgo, Robert, and J. R. Fox. "Maximum weight vectors possess minimal uncertainty." Journal of Physics A: Mathematical and General 10.12 (1977): L233.