Hi @KillMartin
this is indeed a fundamental question deserving proper understanding.
A spectrum plot drawn with the DX*Y option [to be avoided] is intrinsically misleading, since its shape and height will depend on the energy binning (as your second figure demonstrates), while the latter comes from an arbitrary choice that has no vocation to alter a physical quantity representation. Therefore, any histogram plotted as a function of a continuous variable X should always embed the (DX)^-1 term, removing the linear dependence on the bin width.
The linear vs logarithmic alternative for the DE energy binning does not affect the dimension of the resulting quantity, which is always dN/dE. On the other hand, when you plot as a function of logE (on a horizontal logarithmic scale, even independently of the linear/logarithmic choice for DE), you should plot dN/dlogE (Part/cmq/pr) that is [as recalled many times on this forum] E dN/dE (GeV Part/GeV/cmq/pr) = <X> * Y (and not DX * Y !). If you do so, also the spectrum you compare with should of course be plotted on a log E scale as dN/dlogE = E dN/dE (independently of its binning).
As for the integral (sum of dN/dE dE), this is already calculated by FLUKA and printed in the (USRBDX) sum.lis file [this is another piece of information advertised on a regular basis on this forum, we should find the way to do it more effectively].