IMO All of those 3 variables (turbulance, quality of bubble, air intake) are important, obviously they work in conjunction with each other. Having a low turbulance, high air intake skimmer with great bubbles is obvously much better then say having low turbulance with high air intake but crap bubbles. I think the main thing your seeing phi is the improved bubbles as you said. The atman pumps in my oppinion can have a tendancy to "burp" pockets of air, which im sure disturbs the "froth" a bit.
You have to remember you have never tried running a skimmer that is actually rated for your size of tank. So in effect you have always been "overkill" on the air draw of your skimmers. I can say absolutely air draw is important. You obviously need to processs more water in a larger tank, You also need to draw more air to keep the air:water ratio at optimum levels.
These facts are proven by the performance of say..superskimmers for instance..high turbulance and low air saturation kills it. Yea a 220 version processes enouph water to run on a 200gallon, but the air draw and ratio say otherwise. The only time i have ever seen these skimmers work decently is with a recirc mod..even then the high turbulance dampens the function of it.