Ice water for burns does the best.
Converting ice at 32F to water at 32F takes far more heat than simply raising the water or ice by themselves.
It will draw out more heat quicker.
I know this sounds crazy, but when I have a minor burn I immediately run hot water over it—as hot as I can stand. This was told to me by a restaurant employee. I found that the burn rarely blistered when I did this and we owned a restaurant for 40 years.
Don't exactly understand what you were trying to say there at the end. But, indeed:
The enthalpy of fusion is the amount of energy required to convert one mole of solid into liquid. For example, when melting 1 kg of ice (at 0 °C [...]), 333.55 kJ [about 80 (kilo)calories] of energy is absorbed with no temperature change.-Wikipedia
Thus, 1 kg of ice at 32 °F has a far greater cooling power than that same 1 kg of H2O at 32 °F in liquid form. Indeed, merely melting 1 kg of ice - without any temperature increase - would absorb as much heat from the surrounding environment (e.g., the burnt flesh) as would be required to then proceed to elevate the temperature of the resultant meltwater another 80 degrees Celsius (approaching boiling)!
If you want to cool a burn wound, ice would thus be more effective than ice-water - though solid ice would display little to no convection (one of the three ways heat can be transferred). In other words: Liquid water can transfer the heat away from the wound faster.
Light, loose powdered snow would probably be a better choice, but nevertheless, I think that I'd prefer ice-water - since the objective isn't to freeze the damaged tissue, but rather just to remove the excess heat left over from the actual burn event.
Once the burned tissue has been brought back to about 98.6 °F, there's no curative benefit to further cooling. The only benefit further cooling would provide would be to numb the pain (not to be underestimated!).
Regards,