  We present a method for extracting gravitational waves from numerical
spacetimes which generalizes and refines one of the standard methods based on
the Regge--Wheeler--Zerilli perturbation formalism. [abridged] We then present
fully nonlinear three-dimensional numerical evolutions of a distorted
Schwarzschild black hole in Kerr--Schild coordinates with an odd parity
perturbation and analyze the improvement we gain from our generalized wave
extraction, comparing our new method to the standard one. [abridged]
  We find that, even with observers as far out as $R=80 M$--which is larger
than what is commonly used in state-of-the-art simulations--the assumption in
the standard method that the background is close to having Schwarzschild-like
coordinates increases the error in the extracted waveforms considerably. Even
for our coarsest resolutions, our new method decreases the error by between one
and two orders of magnitudes. Furthermore, we explicitly see that the errors in
the extracted waveforms obtained by the standard method do not converge to zero
with increasing resolution. [abridged]
  In a general scenario, for example a collision of compact objects, there is
no precise definition of gravitational radiation at a finite distance, and
gravitational wave extraction methods at such distances are thus inherently
approximate. The results of this paper bring up the possibility that different
choices in the wave extraction procedure at a fixed and finite distance may
result in relative differences in the waveforms which are actually larger than
the numerical errors in the solution.
