What does "each pathway define an independent genotype" mean?
This is somewhat outside my field, but basically it seems they're defining the independence of the three hypothetical metabolic pathways in two different ways:


Mathematically the pathways are not independent since P₃ is a combination of P₁ and P₂. 
Biologically the pathways are independent since each requires a different combination of enzymes and therefore genes (the genotype). This is the method used in MPS. 


Does that answer your question?