I am trying to understand the behavior of a pulse
To me, your LTspice (blue) model seems not to be far away from the real data (circles). If you closely look at the real data, you see a sine component with approx. 30ns period whereas your model seems to have about 5-10ns period. 

So you might want to increase the Cs or Ls in your model in order to get a lower resonance frequency and to get a better match.