Is there any way to identify ...

Is there any way to identify which type of phosphorus (P) is exported from agricultural watersheds?

Yes. Analyse the water leaving watershed for the component parts of total P. i.e. phosphate-P (often called dissolved reactive P [DRP] or soluble filterable P), dissolved organic P, particulate P and total P [TP].  Total P gives the maximum P in the sample, particulate P gives the solids component and DRP gives the inorganic component. The dissolved organic P component can be analysed as for total P after filtration and subtracting the DRP. These results will give you the form of the P leaving the watershed.

Note that DRP is rapidly bound to iron particles in the soil / sediment or may be assimilated into aquatic plants including algae. Consequently, there may not be any (or perhaps 10) then filtering before analysing the DRP content using the TP method.

You may be able to distinguish between chemical fertilisers and organic matter (compost and manure) by analysing the oxygen-18 isotope stable isotopic signature of the DRP once it is extracted.