You'll generate a ton of dissonance on here by asking about cluchless shifting, you're going to get about as many opinions as if you'd asked "what's the best kind of oil," or "what's a TRE do?"
Personally, I use a mix of both. Probably 8 times out of 10 I'll just blip the clutch lever for a fraction of a second when I shift, sort of like thebbbusa is talking about above. If you're going to shift without clutching you need to unload the transmission first or you're going to round the dogs off, especially in the lower gears like one and two. Just click the throttle closed for a microsecond at the same instant you apply upward pressure on the shift lever and then get back into the throttle.
Downshifting clutchlessly is probably a tad harder on parts than upshifting and in my mind there isn't a lot of reason to do it because I like to clutch and blip the throttle as I downshift to bring the rpms up slightly and keep the bike in the powerband. Be careful if using this technique with your solid clutch as the chances of locking up the rear wheel increases greatly when you lose your B.T.L. clutch. *chirp* *chirp*
Your airshifter works the same way as the clutchless upshifting I described above except when you press the button it uses an electronic device to cut the ignition (or kill fuel delivery in the case of the Muzzy) for a fraction of a second at the same instant as the solenoid delivers air pressure to the ram to move the shift lever. It's the same thing, it just happens a lot faster than you could do it manually, and you can wring the throttle open and just click the button instead of rolling on and off the gas for every gear.