gistfile1.txt

I've enabled normal logging and RDP'd into the instance. You can find the build report here: https://ci.appveyor.com/project/Vannevelj/vsdiagnostics/build/1.8.627

When I look at the folder structure at the end of the build, I can see that is laid out like this:

\packages\VSDiagnostics\1.9.3\
 \lib\
  \portable\
   VSdiagnostics.dll
 \tools\
  install.ps1
  uninstall.ps1
 VSDiagnostics.1.9.3.nupkg

This is the reason it can't find the analyzer under \packages\VSdiagnostics.1.9.3\tools\analyzers\VSDiagnostics.dll

However this is a surprise an sich because if I create a standalone application on my own machine it has this folder structure:

\packages\VSDiagnostics\1.9.3
 \tools\
  \analyzers\
   VSDiagnostics.dll
  install.ps1
  uninstall.ps1
 VSDiagnostics.1.9.3.nupkg

You can verify this for yourself by creating a new console application and adding VSDiagnostics 1.9.3 from NuGet.
Does this give you any idea as to what might be the cause?