What allows all 3 states of H₂O to be present at the same time?
At the triple-point of a substance, all three phases can be present. Since it takes additional energy to go from ice at 0 C to liquid water at 0 C, there can be a considerable overlap for the two substances, e.g. 1 g ice with 99 g water or 99 g ice with 1 g water, but for thermodynamic equilibrium, the triple point is very narrow.

BTW, the first well-know American chemist, Josiah Willard Gibbs, pioneered in defining the relationship of temperature, pressure and state. See  Gibbs' phase rule for more information.