The dynamics and the decay processes of inner-shell excited atoms are of great interest in physics, chemistry, biology, and technology. The highly excited state decays very quickly through different channels, both radiative and non-radiative. It is therefore a long-standing goal to study such dynamics directly in the time domain. Using few-cycle infrared laser pulses, we investigated the excitation and ionization of inner-shell electrons through laser-induced electron re-collision with the original parent ions and measured the dependence of the emitted x-ray spectra on the intensity and ellipticity of the driving laser. These directly re-colliding electrons can be used as the initiating pump step in pump/probe experiments for studying core-hole dynamics at their natural temporal scale. In our experiment we found that the dependence of the x-ray emission spectrum on the laser intensity and polarization state varies distinctly for the two kinds of atomic systems. Relying on our data and numerical simulations, we explain this behavior by the presence of different excitation mechanisms that are contributing in different ratios to the respective overall x-ray emission yields. Direct re-collision excitation competes with indirect collisions with neighboring atoms by electrons having &\#x201C;drifted away&\#x201D; from the original parent ion.