Scalar curvature for a given metric does not fit the expected sign and numerical factor

Question

For a metric given by the infinitesimal length element \begin{equation} \label{Special_metric} {\rm d}s^2=(r/r_{0})^4~c^2{\rm d}t^2-{\rm d}r^2-r^2{\rm d}\Omega^2~,\tag{1} \end{equation} where $r$ is radius of sectional curvature, $r_{0}$ some reference value, $c$ light velocity, and ${\rm d}\Omega$ an infinitesimal surface element on the unit sphere, the calculated scalar curvature, $S=g^{ik}R_{ik}$, results in \begin{equation} \label{Ricci_scalar_curvature} S=\frac{12}{r^2},\tag{2} \end{equation} This expression is the same as for the scalar curvature of a 4-sphere in Euclidean space, $S=4~(4-1)/r^2$, as described in https://en.wikipedia.org/wiki/Scalar_curvature. However, the metric (1) describes a 3-dimensional hyperbolic space that can be identified with a local subset of (3+1)-dimensional Minkowski space (wiki again). The corresponding scalar curvature in that case would be equal \begin{equation} \label{Ricci_scalar_curvature 2} S=-\frac{3~(3-1)}{r^2}=-\frac{6}{r^2}.\tag{3} \end{equation} Where am I making a mistake?