The second paragraph in the question describes two solutions separated by a semipermeable membrane (a membrane that allows water through, but not anything dissolved in the water). As it says, in those conditions, water will move from the section with low osmotic pressure to the solution with high osmotic pressure.
Osmotic pressure, like other colligative properties, is proportional to the concentration of solutes in a solution. So you can think of it is a measure of how much other stuff is in the water. Low osmotic pressure means low solute concentration. High osmotic pressure means high solute concentration. Water moving across a semipermeable membrane from an area of low solute concentration to an area of high solute concentration may be a little more intuitive. Given a barrier to solutes but not water, water will move in a way that will even out the concentration.
Now come back to the definition of osmotic pressure. The definition starts by setting up a very special set of conditions to allow us to measure the pressure -- pure solvent in one compartment, a known concentration in another compartment, and the key difference, pressure to prevent water from moving the way it otherwise would.