  We simplify and complete the construction of fully $O(D)$-equivariant fuzzy
spheres $S^d_L$, for all dimensions $d\equiv D-1$, initiated in [G. Fiore, F.
Pisacane, J. Geom. Phys. 132 (2018), 423]. This is based on imposing a suitable
energy cutoff on a quantum particle in $\mathbb{R}^D$ in a confining potential
well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff
and the depth of the well diverge with $L\in\mathbb{N}$. As a result, the
noncommutative Cartesian coordinates $\overline{x}^i$ generate the whole
algebra of observables $A_L$ on the Hilbert space $H_L$; $H_L$ can be recovered
applying polynomials in the $\overline{x}^i$ to any of its elements. The
commutators of the $\overline{x}^i$ are proportional to the angular momentum
components, as in Snyder noncommutative spaces. $H_L$, as carrier space of a
reducible representation of $O(D)$, is isomorphic to the space of harmonic
homogeneous polynomials of degree $L$ in the Cartesian coordinates of
(commutative) $\mathbb{R}^{D+1}$, which carries an irreducible representation
${\bf\pi}_L$ of $O(D+1)\supset O(D)$. Moreover, $A_L$ is isomorphic to
${\bf\pi}_L\left(Uso(D+1)\right)$. We resp. interpret
$\{H_L\}_{L\in\mathbb{N}}$, $\{A_L\}_{L\in\mathbb{N}}$ as fuzzy deformations of
the space $H_s:={\cal L}^2(S^d)$ of (square integrable) functions on $S^d$ and
of the associated algebra $A_s$ of observables, because they resp. go to
$H_s,A_s$ as $L$ diverges (with $\hbar$ fixed). With suitable
$\hbar=\hbar(L)\stackrel{L\to\infty}{\longrightarrow} 0$, in the same limit
$A_L$ goes to the (algebra of functions on the) Poisson manifold $T^*S^d$; more
formally, $\{A_L\}_{L\in\mathbb{N}}$ yields a fuzzy quantization of a coadjoint
orbit of $O(D+1)$ that goes to the classical phase space $T^*S^d$.
