The raw area and the background area are calculated in FoundPeak.cs:
https://github.com/ProteoWizard/pwiz/blob/master/pwiz_tools/Shared/Common/PeakFinding/FoundPeak.cs
float rawArea = GetAreaUnderCurve(intensitiesToSum);
float backgroundArea = GetAreaUnderCurve(intensitiesToSum.Select(intensity => Math.Min(backgroundLevel, intensity)));
and then the transition peak area that Skyline reports is the rawArea minus the backgroundArea.
Note that when you are calculating the integral, the only difference that rectangles versus trapezoids makes is in how the two points at the ends of the range are treated. The way the internal points along the integral get summed is the same for trapezoids and rectangles. In the implementation of "GetAreaUnderCurve", the first and last intensities get divided by 2, which I think means that it's the equivalent to using trapezoids.
I do not think there's anything incorrect in the new picture, but I'm not sure that I understand what the issue might be.