With an open pan, most of the heat loss is due to evaporation and convection from the top surface, and unless you have a very odd shaped pan then the open surface area is constant no matter how much water you have in the pan. Hence the larger temperature differential when adding cold water later results in more heat loss. The surface area available for heat loss directly from the pan will actually be greater with less water so again more heat loss with the adding water later method.
With a closed pan, the external surface area for heat loss is constant no matter how much water you have in the pan, hence the larger temperature differential when adding cold water later results in more heat loss.
For all cases I’m assuming the temperature of the pan itself is constant across the pan, but that’s not a bad assumption for a metal pan which conducts heat well.
In summary the quickest way to boil the water and the most efficient is to put all the water in the pan at the start.