Agreed that different sites will provide different definitions for the term power shifting. However, shifting while not pressing the clutch is "NOT" fast and quick since you have to wait for the rpms to drop to the "SWEET" spot. Changing gears without hitting the clutch is not fast, and is not done by professionals in racing. The racers want to maintain the rpms as high as possible to always be around their peak horsepower range. Power shifting (the one where you don't take your foot off of the gas) provides faster than normal shifting, and higher speeds, hence the term "power". Changing gears without hitting the clutch requires you to take your foot off of the gas a bit. It is good to know how to shift without hitting the clutch in the event of a clutch failure.
And power shifting is bad...very bad. It is as effective in new cars just as it is in older cars, if not more now, because in the event of tranny failure, the car is under warranty and can be replaced free. The difference between the cars that people used to power shift in the muscle car era, and the cars that we all have today is that the older cars had globs of power, where we don't. If you slap on a turbo to one of the four bangers, change the internals, and do other stuff to bring up the horsepower and torque to about the same range as the muscle cars, then power shifting will show you a better increase in acceleration than the muscle cars can due to the lighter weight of our cars. However, the tranny wont hold as long as one of those muscle car tranny's because it just wasn't designed to take the abuse. Something will break, eventually.
By the way, I did a seach as well, and found that most of the websites agree that power shifting is when you don't take your foot off the gas. Usually found in racing sites and muscle car forum boards. But I also found some that said it was when you didn't use the clutch.