What is difference between running as admin and having admin rights?
99% of users use accounts with admin rights, it just FAF does not need them and does not ask for them.
But you can still "run as admin" and that will trigger UAC dialog window apper...
My suspection is that there is something different with windows install or settings, for example UAC is off or something... not really sure at all...
or opposite... if when you run FAF if UAC appears you need to say "no" or turn it off or adjust file settings so it does not run with admin rights
(you can force application to run without admin rights even if it wants to, but afaik it needs registy editing, it is not hard i can provide instructions if you need)