Supercooling, freezing point depression, and finding the right freezing point
Melting (and boiling) points are only defined for pure substances. Rather than defining them as ‘the temperature when pure substance $\ce{X}$ melts/boils’ they are better defined as:

The melting point is the temperature at which for a certain defined pressure (usually $1~\mathrm{bar} = 10^5~\mathrm{Pa}$) the liquid and solid phases of a pure substance are at equilibrium.
The boiling point is the temperature at which for a certain defined pressure (usually $1~\mathrm{bar} = 10^5~\mathrm{Pa}$) the liquid and gaseous phases of a pure substance are at equilibrium.

(I didn’t copy that from anywhere, I made it up. Apologies to everyone who I was unintentionally quoting without attribuition.)
Thus, you can supercool pure liquids but if you allow the supercooled liquid to reach equilibrium at its supercool temperature (and standard pressure), it will solidify. Similarly, you can overheat solids but if you let the overheated solid reach equilibrium at the overheated temperature, it will liquidify. Thus, neither the supercool liquid’s nor the overheated solid’s temperature can be considered the melting point, because neither phase is stable at equilibrium.