Can you store them with the probe immersed in distilled water?
No. The probe is filled with a ion-saturated (typically potassium) solution or gel. Its important to keep that solution saturated with that ion. The glass probe is actually permeable. Immersing the probe in a solution with very low ionic strength creates an osmotic pressure that draws those ions out of the probe and into the exterior solution. You end up depleting the ion solution inside the probe. pH probe storage solutions are typically high in the particular ion and when the probe is immersed in that solution, then there is not the osmotic stress drawing ions out of the probe. The probe solution stays saturated.