  Theoretical predictions for lepton-induced single-pion production (SPP) on
$^{12}$C are revisited in order to assess the effect of different treatments of
the current operator. On one hand we have the asymptotic approximation, which
consists in replacing the particle four-vectors that enter in the operator by
their asymptotic values, i.e., their values out of the nucleus. On the other
hand we have the full calculation, which is a more accurate approach to the
problem. We also compare with results in which the final nucleon is described
by a relativistic plane wave, to rate the effect of the nucleon distortion. The
study is performed for several lepton kinematics, reproducing inclusive and
semi-inclusive cross sections belonging to the low-$Q^2$ region (between 0.05
and 1 GeV$^2$), which is of special interest in charged-current (CC)
neutrino-nucleus 1$\pi$ production. Inclusive electron results are compared
with experimental data. We find non-trivial corrections comparable in size with
the effect of the nucleon distortion, namely, corrections up to 6\%, either
increasing or diminishing the asymptotic prediction, and a shift of the
distributions towards higher energy transfer. For the semi-inclusive cross
sections, we observe the correction to be prominent mainly at low values of the
outgoing nucleon kinetic energy. Finally, for CC neutrino-induced 1$\pi^+$
production, we find a reduction at low-$Q^2$ with respect to both the
plane-wave approach and the asymptotic case.
