I follow the line that pressing 'soft' is definitely misleading. It is the depth of pressed willow that is important to a bat's response. As a general rule, a deeper pressing will improve durability, but reduce ball speed off the bat. A shallow press could provide what players experience as a 'trampoline' effect. But this of course depends on the quality of the cleft, and therefore a uniform ' shallow' press will not be optimum for all bats. Bat makers who still press each cleft on its merits will be close to optimising performance for every bat.
The trampoline effect occurs when the fibre/cell compression from impact recovers while the ball is still in contact with the bat, but on its exit path. Being able to tune that recovery to the contact time is what you ideally would try to engineer. If the pressed layer is deeper, then you have a stiffer layer that will deform less, and so return less energy to the ball, although paradoxically it will also absorb less energy from the ball during impact. It's very much swings and roundabouts, which makes 'tuning' a bat for optimum performance a complex process to get right repeatably
However, some perspective is also required, since the gains that could be made from pressing may be marginal compared to the influence of bat mass, bat speed, and technique. Hence we see some pro's (such as KP) stating that they are indifferent to the minute technicalities of their bats. If it feels right, and its coming from a trusted bat maker, then that's good enough. It doesn't stop other pros doing the opposite, and trying every trick possible to squeeze every ounce of performance from the bat. I've heard that Herschelle Gibbs for a while experimented with unpressed bats, which flew, but also broke at an alarming rate. I've also heard he experimented with many other things as well, but that's another story...