= Solution
<Linearization> removes the <Poisson bracket> from $\nabla_\parallel$. For Fourier perturbations proportional to $e^{i\mathbf k\cdot\mathbf x-i\omega t}$, the two reduced equations become
$$
-i\omega\Psi=iv_A^2d_ik_\parallel b,\qquad-i\omega b=id_ik_\parallel k_\perp^2\Psi.
$$
Eliminating either amplitude gives
$$
\boxed{\omega^2=v_A^2d_i^2k_\parallel^2k_\perp^2,\qquad\omega=\pm k_\parallel v_Ad_ik_\perp.}
$$
Since $k=(k_\perp^2+k_\parallel^2)^{1/2}=k_\perp[1+O(\epsilon^2)]$, this is the specified <dispersion relation> to the accuracy of the anisotropic reduction. The reduced equations cannot retain the subleading correction replacing $k_\perp$ by the full $k$.
The full <electron magnetohydrodynamics> equation checks that correction directly. Its <linearization> is $\partial_t\delta\mathbf B=-\alpha B_0\partial_z\nabla\times\delta\mathbf B$. Hence
$$
-i\omega\delta\mathbf B=\alpha B_0k_\parallel\mathbf k\times\delta\mathbf B.
$$
Applying the cross-product operator twice and using $\mathbf k\cdot\delta\mathbf B=0$ gives
$$
\boxed{\omega^2=(\alpha B_0)^2k_\parallel^2k^2=v_A^2d_i^2k_\parallel^2k^2.}
$$
Thus the full model gives the stated relation exactly and the reduced model gives its leading anisotropic limit, with all signs and normalization factors consistent.
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