The reason that a lawn irrigated with rainwater looks better after verses tap or groundwater is that Nitrogen is very mobile and changes form readily from a gaseous state to being very water soluble being carried by rainwater. A good rain always contains Nitrogen(N) which is taken up by the lawn's root zone. I'm not sure how much but that's likely one reason any given lawn looks better after a good rain. Additionally, I also recall vaugely something about ion exchange with rainwater but can't remember anything else regarding that. Also that with soil testing, N is not measured since it's so mobile. .