How do you calc this exactly? I mean, the calculation page does seem to give the formula, but for example
this calc seems to do something different and it was accepted. Is it just another form of the formula in question? Or is it because it was different back then?
With temperature change calcs, there are two things you need to consider, which may or may not be applicable. One is the actual energy needed to change an object's temperature from one level to another. To determine this, you need three things: the mass of the object, the heat capacity (a constant physical property which expresses how much energy is needed to change the temperature of a kg of substance by a certain amount), and the change in temperature in Kelvin. That can all be expressed in this equation.
Energy = Object Mass * Heat Capacity (J/kg*K) * (Final Temp - Initial Temp)
The other thing that you need to consider is what's called the latent heat, or the energy actually needed to change an object from one physical state of matter to another. The units for this can vary, since there different thermodynamic values for things like melting (fusion), or vaporization. Normally though, these values are expressed in a certain amount of Joules over the quantity of a substance (kJ/kg, J/g, J/cm^3, etc.) You then just multiply that by whatever quantity you have, then add it to your result.
Now, looking at the calc you linked, I can't really tell what's being done exactly, but it does seem to fundamentally be the same idea. It's just expressed algebraically in a way I'm having trouble comprehending. Freezing calcs are also just generally weird because they involve removing energy from a system rather than adding it in, basically meaning you are working with a net negative amount of energy if you did it the way I did above (the initial temp would be higher than the final temp, causing a net negative amount of energy). In which case, they might've just written it in a way where they wouldn't have had to correct negative numbers