The old conventional wisdom for slalom turns was "quick on and off the edges." While some may think that "conventional" has very little to do with slalom racing these days, the advice still holds!
In classic slalom turns, the racer would turn the skis to virtually the direction they needed to point at the end of the turn, then VERY briefly set and release the edge of the downhill ski to accomplish the direction change. They spent as little time on their edges (what you are calling the "resistance phase")as possible. Skis had very little sidecut (hourglass shape) to enable this quick edgeset, so they did not turn much when the edges were engaged.
One common misconception about classic slalom technique is that "quick on and off the edges" must mean EARLY on the edges. It doesn't! If the edgeset is to be as brief as possible, it must also come as late in the turn as possible. So, quick as they were, slalom turns required a fair amount of patience to steer the skis while unweighted, then set the edge quickly, once the direction is attained.
Today's slalom skis are very different, of course. 155cm (for men, 150cm for women), with very deep sidecuts, allows the skis to actually carve the tight turns of slalom. In theory, at least, racers can engage the edges early in the turn and "ride" the skis around the arc, more like a giant slalom turn.
But watch Bode Miller, especially, although most of the men demonstrate the technique well. Bode spends VERY little time on his edges, usually. His technique is not that different from the classic slalom technique in this respect. While his skis do carve a tight arc when he engages the edges, that engagement is still very brief, and late in the turn.
The World Cup women do the same thing, but typically much less obviously and to less of an extreme. They often roll from edge to edge, riding their extreme sidecut skis from arc to arc. But the men illustrate your coach's advice quite clearly--as straight a line as possible, with as brief an edge engagement (and tight a turn) as they and their skis can handle.
On the other hand, overdoing this quick, late edgeset can cause a braking action, rather than a clean and fast direction change. Experiment and find out just what you are capable of!
It sounds to me like your coach is pretty enlightened.