> Basically as it gets colder water exchanges energy with the environment and gets colder.

> But once it reaches freezing temperature, it can no longer get colder and all the energy is used for the formation of crystals

Water at freezing temperature can get much colder without freezing. https://en.wikipedia.org/wiki/Supercooling:

“Water normally freezes at 273.15 K (0.0 °C; 32 °F), but it can be "supercooled" at standard pressure down to its crystal homogeneous nucleation at almost 224.8 K (−48.3 °C; −55.0 °F).”