  Quantum or classical mechanical systems symmetric under $SU(2)$ are called
spin systems. A $SU(2)$-equivariant map from $(n+1)$-square matrices to
functions on the $2$-sphere S^2, satisfying some basic properties, is called a
spin-$j$ symbol correspondence ($n = 2j \in \mathbb{N}$). Given a spin-$j$
symbol correspondence, the matrix algebra induces a twisted $j$-algebra of
symbols. In the first part of this paper, we establish a more intuitive
criterion for when the Poisson algebra of smooth functions on $S^2$ emerges
asymptotically ($n \to \infty$) from the sequence of twisted $j$-algebras. This
more geometric criterion, which in many cases is equivalent to the numerical
criterion obtained in [20] for describing symbol correspondence sequences of
(anti-)Poisson type, is now given in terms of a classical (asymptotic)
localization of symbols of all projectors (quantum pure states) in a certain
family. For some important kinds of symbol correspondence sequences, such a
classical localization condition is equivalent to asymptotic emergence of the
Poisson algebra. But in general, the classical localization condition is
stronger than Poisson emergence. We thus also consider some weaker notions of
asymptotic localization of projector-symbols. In the second part of this paper,
for each sequence of symbol correspondences of (anti-)Poisson type, we define
the sequential quantization of a smooth function on $S^2$ and its asymptotic
operator acting on a ground Hilbert space. Then, after presenting some concrete
examples of these constructions, we obtain some relations between asymptotic
localization of a symbol correspondence sequence and the asymptotics of its
sequential quantization of smooth functions on $S^2$.
