When you are solving the Einstein field equations (EFE), you basically have to input a stress–energy tensor and solve for the metric.
$$ R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} = 8 \pi T_{\mu \nu} $$
For a vacuum solution we have:
$$ T_{\mu \nu} = 0 $$ This yields: $$ R_{\mu \nu} = 0 $$
This means that the local curvature of an inertial frame of reference is zero.
But, setting the stress–energy tensor to zero, could be given in multiple situations: In flat spacetime, around a non-rotating black hole, around a planet, etc.
When I read about this equations in divulgation books, they portrait the Einstein field equations as:
$$ \text {Space-Time Curvature} = \text{Energy} $$
But with this interpretation, by setting $T_{\mu \nu} = 0$, you are saying that the energy is zero, hence no curvature, but you are able to get more solutions than the Minkowski metric (which is the only solution with truly no energy and with no curvature).
Are this books interpretations wrong or is there something I'm not getting from the true meaning of the equation? How would you distinguish, while solving the EFE, from a truly flat spacetime, from a locally flat spacetime?