How may the water from rain be tested to determine it is natural, or from cloud seeding?
The AgI would act as a cloud condensation nuclei (CCN) on which water vapour would condense allowing cloud droplet formation. Presumably, you could use scanning electron microscopy to determine the composition of the CCN and determine whether cloud seeding has taken place.

Apparently, his has already been done with cloud chamber (so, artificial scenario) measurements (Zimmerman et al., 2007 - "Environmental scanning electron microscopy (ESEM) as a new technique to determine the ice nucleation capability of individual atmospheric aerosol particles").

However, considering that there has been ESEM measurements in field situations before (Pöschl et al., 2010 - "Rainforest Aerosols as Biogenic Nuclei of Clouds and Precipitation in the Amazon") to determine CCN composition, I suppose a similar set up could be performed.