An earthquake on the ocean floor produces a tsunami that hits a remote island. Is the water that hits the island the same water that was above the earthquake on the ocean floor?
I am a bit out of my turf here but I would say that, no it is not the same water.
The energy generated during the earthquake will appear as a sudden movement of the seabed; this in turn will act as an oscillator moving the water above it vertically (like when you throw a stone in a lake) producing waves propagating outward from the epicenter BUT not changing the horizontal position of the water molecules.
It is possible that near the coast in shallow waters, the sea water starts to roll instead of waving up and down only.
As I said I am a bit out of my area so I'll also try to call in for the experts!