All first order phase transitions have a change of volume. With different pressures you need to consider the sign of the work $P\Delta V$ that needs to occur during the phase change. If $\Delta V$ is positive, the phase change will occur at a higher temperature for higher pressure. If negative, the phase change will occur at a lower temperature.
(Note that how the temperature is changed, or how fast, has nothing whatsoever to do with thermodynamics - that is a kinetic issue and does not impact the relative free energies of the various phases.)
Now, for boiling water, the molar volume of steam is larger (by a lot) than the molar volume of water at the boiling point. Increasing the pressure results in higher boiling points. This is the basis of pressure cookers, superheat steam engines, etc. On the other hand, ice has a lower molar volume than water (it floats), so increasing pressure leads to a freezing point decrease.