you're right - we have been on either side of a coin talking about two different things! However, you normally sail downwind with your boards up right? It's the same reasoning, the slip helps you get from point A to a downwind B faster*. No additional water will flow under the boat than if you were sailiing downwind, boards up, in a stationary body of water.
I was talking about another example and the site link that Jamie provided is a perfect example...modify it a little and think about an airplane: Have you ever flown on an airplane? You're at 10,000 feet and flying at 500 mph in a straight line yet you can walk around and feel no additional forces. Your boat and the current is the same thing. Your boat (you on the plane) does not feel the current moving - it only knows it's sailing as usual. The same reasoning applies to us in that we don't have to modify the way we move because the earth is spinning at 1000mph - we don't feel anything because we're on it.
*there are exceptions but that would be a whole nother' topic.