Neither or both, depending on your point of view. Refer to the illustration: the surfaces with $\eta$ held constant are oblate spheriods. From the equations of the transformation to Cartesian coordinates, these spheroids have the parameterization $$\begin{align}x &= (a\cosh\eta)\sin\theta\cos\psi \\\ y &= (a\cosh\eta)\sin\theta\sin\psi \\\ z &= (a\sinh\eta)\cos\theta.\end{align}$$ I’ve added parentheses to emphasize the constant quantity in each equation. The half-axis lengths of the spheroid are therefore $a\cosh\eta$ and $a\sinh\eta$. Similarly, the surfaces with constant $\theta$ are hyperboloids of revolution with half-axis lengths $a\cos\theta$ and $a\sin\theta$. The angle $\theta$ represents the aperture half-angle of the hyperboloid’s asymptotic cone. Surfaces with constant $\psi$ are half-planes that make an angle of $\psi$ with the positive $x$-$z$ half-plane.