They're empirical based on load tests largely in the US 50-70 years ago (knowledge of which - that it's all 'this is empirical curve fitting to load tests done with construction equipment and techniques in a certain types of soil conditions in certain regions of the United States from the 50s to the 70s - seems to have been lost with the advent of magical computer programs that hide all of this). So it relates to the conditions of the original tests, one is for tests where the p-y curve is believed to be governed by stiff clay with water above the ground surface or at least at you think water would fill the soil-pile gap when it deflects. The latter for the case where you don't think water will fill the soil-pile gap. I.e. those were the conditions in the tests used to develop the p-y curve.
With free water implies there is water, and that the water has the ability to move freely. This will cause piping of soil due to pier movements, and create a gap between the soil and the pier. Therefore, there is a loss of support, and an increase in deflection verses a pier without free water. The key factor is the water has to be able to freely move.
