Spin-$n$ particle comes back to itself after $360/n$ degree rotation

On the Wikipedia page for spin, a claim is made that a spin-$n$ particle comes back to itself after a $360/n$ degree rotation.

I quote:

A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180 degrees can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180 degrees and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.

My question is: Is this claim true? If it is false, what is the correct claim?

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I believe the claim is false for $n>1$. I believe that for half-integer spin particles, no less than a $720$ degree rotation is required, and for integer spin no less than a $360$ degree rotation is required to bring the internal state back to itself. To explain my reasoning, consider the generators of rotation about the z-axis:

$S_z^{1/2} = begin{pmatrix}
1/2 & 0
0 & -1/2
end{pmatrix}$

$S_z^{1} = begin{pmatrix}
1 & 0 & 0
0 & 0 & 0
0 & 0 & -1
end{pmatrix}$

$S_z^{3/2} = begin{pmatrix}
3/2 & 0 & 0 & 0
0 & 1/2 & 0 & 0
0 & 0 & -1/2 & 0
0 & 0 & 0 & -3/2
end{pmatrix}$

$S_z^{2} = begin{pmatrix}
2 & 0 & 0 & 0 & 0
0 & 1 & 0 & 0 & 0
0 & 0 & 0 & 0 & 0
0 & 0 & 0 & -1 & 0
0 & 0 & 0 & 0 & -2
end{pmatrix}$

A rotation about the $z$-axis with angle $theta$ is given by $e^{iS_z theta}$.
Since the rotation matrices are diagonal, we have

$e^{iS_z^{1/2}theta} = begin{pmatrix}
e^{itheta/2} & 0
0 & e^{-itheta/2}
end{pmatrix}$

$e^{iS_z^{1}theta} = begin{pmatrix}
e^{itheta} & 0 & 0
0 & 1 & 0
0 & 0 & e^{-itheta/2}
end{pmatrix}$

$e^{iS_z^{3/2}theta} = begin{pmatrix}
e^{i3theta/2} & 0 & 0 & 0
0 & e^{itheta/2}& 0 & 0
0 & 0 & e^{-itheta/2} & 0
0 & 0 & 0 & e^{-i3theta/2}
end{pmatrix}$

$e^{iS_z^{2}theta} = begin{pmatrix}
e^{i2theta} & 0 & 0 & 0 & 0
0 & e^{itheta} & 0 & 0 & 0
0 & 0 & 1 & 0 & 0
0 & 0 & 0 & e^{-itheta} & 0
0 & 0 & 0 & 0 & e^{-i2theta}
end{pmatrix}$

Clearly, we need $theta = 4pi$ for spin-$1/2$ and $theta = 2pi$ for spin-$1$ as the minimum nonzero rotation angle in order to get back to the identity matrix.

However, because $1/2$ is an element on the diagonal of $S^{3/2}_z$, we also need $theta = 4pi$ as the minimum nonzero rotation angle to get back to the identity matrix, not $theta = 4pi/3$. For spin-$2$, we need $theta = 2pi$ and not $theta = pi$ as minimum nonzero rotation angle in order to get back to the identity matrix.

Quite generally, the above seems to say that half-integer spins require a minimum rotation angle to "get back to themselves" of $theta = 4pi$, while (nonzero) integer spins require $theta = 2pi$.

The claim that the rotation angle is $theta= 2pi/n$ for spin-$n$ particles appears to be incorrect.

Though I feel mostly convinced by my arguments above, I have heard the Wikipedia page’s claim more than once before, so I fear I am missing something, hence the motivation for my question.