Rain can add nitrates (another algae nutrient), but I don't believe it adds phosphates.

That said, I have not fully explored the avenues by which phosphates enter the pool. However, I'm sure the two primary routes are treated fill water (well water wouldn't normally contain phosphates, but 'city' water usually does) and phosphonate based metal / stain control products. A third route are United Chemical's products. Many of those contain phosphates, apparently as a detergent to improve chlorine & bromine penetration of algal biofilms.

The issue with city water is the reason we'll need to test BOTH your pool water AND your fill water.