Why does a high osmotic pressure pull water in, yet a high hydrostatic pressure pushes water out?
Great question, the answer has to do with the definition of osmotic pressure and the difference between this pressure and "hydrostatic pressure" which is the water pressure you are thinking of.

Here is the definition of "osmotic pressure" given by Google:


  the pressure that would have to be applied to a pure solvent to prevent it from passing into a given solution by osmosis, often used to express the concentration of the solution.


Osmotic pressure in this definition is best illustrated by the commonly used tool to show osmotic pressure, often seen in a textbook as something like this:



In this picture, osmotic pressure is roughly how hard do you have to push down on the water on the right to make the levels the same on the left and right, despite the difference in concentrations. Osmotic pressure is not a pressure of water, as your question seems to imply, it is a pressure of the solutes in the water.

If it helps, I think it would be appropriate to think of this as sort of a "suction" on the water. Net flow of water stops when:

osmotic pressure - hydrostatic pressure = 0

...on the right side of the diagram.