I think a lot of your confusion here is due to the fact that "flatter" is being used to describe different parts of sail shape: depth, angle of attack and twist.
how much "shape" the sail has. this is the thick vs thin wing that Nick described so well- thicker wing (more depth)=more power, thin wing (less depth)=less power. you can see how much depth the sail has by imagining a straight line from the front edge of a batten to the back edge; how far off the actual sail is from that imaginary straight line is is the depth. Your primary controls for this are downhaul, outhaul (not as effective as the other two) and mainsheet. when you tighten any of these controls, they stretch out the sailcloth and bend the mast, which gives you less depth in the sail and therefore less power. this is what MMadge meant when she said a flatter sail keeps you in control better.
Angle of Attack:
This is the angle that the sail makes with the wind direction:
Up to a point, higher angles of attack (bigger difference between the wind direction and where your sail is pointed), give you more power. However, beyond a certain point, the sail stalls and gives you no power and a lot of drag. Too low of an angle of attack and the sail just faces into the wind and doesn't produce power. This is controlled by the combination of sheet tension (see "twist") and traveller position for a given sheet tension, having the traveller towards the center of the boat increases the angle of attack and the amount of power).
If you imagine looking down on the boat from above while sailing upwind, the sail isn't all going to be pointed in the exact same direction. the boom will be closer to the centerline of the boat than the upper part of the sail, which will be falling off to leeward. The angle between the two is the "twist" in the sail. its controlled by the tension of the sheet. in higher winds, there is more force trying to push the top of the sail to leeward, so you need more mainsheet tension to maintain the same amount of twist. the correct amount of twist will give you the "optimum" angle of attack for the sail from the boom all the way up to the top of the mast.
Putting it all together:
in light to moderate conditions, where you're not overpowered (tipping too much), you want the traveller on centerline and sheet moderately tight (to keep the optimum amount of twist in the sail). If you're temporarily overpowered in a gust, you can ease the sheet, which will allow the top of the sail to twist off more, reducing the angle of attack towards the top and producing less power. when the gust is past, you can bring the sheet back in to power the top of teh sail back up.
However, as the wind builds for longer periods, you want to depower by decreasing the depth of the sail. dropping the traveller will reduce the angle of attack of the whole sail, depowering it, and allow you to carry more sheet tension without stalling the top part of the sail.
Off the wind the same basic principles apply, but you let the traveller out much further so you can sheet in some and decrease the amount of twist in the top of the sail.
Clear as mud, right?? The best way to see all of thise is to go out with someone who really knows Hobie Cats, and can drive while making various adjustments so you can watch the effect they have.
As for your friend's "certification", I've generally found that they aren't necessarilly worth anything. I know some really good sailors who have no formal training, and some terrible sailors who are "certified".