Canon flash logic remains somewhat mysterious to me but I too think of flash in P mode as yielding a "flash" photo whereas flash in Av mode yields a "fill" effect. This would apply to the G12 as well as DSLR; however, it's really not that simple. The general idea is that, with flash, P mode shutter speeds begin at 1/60 and go up to max sync speed. Av mode shutter speeds begin at slower shutter speeds (if necessary for ambient exposure) and go up to max sync speed. But this generality is not always true.
For the G12, in order to get the slow shutter speed ( < 1/60) with flash, you'll have to set the flash control for "slow syncro". It turns out that P *and* Av modes can be set for slow syncro. So P and Av could both produce "fill" flash although I wonder if the actual algorithms differ.
My 7D has three options for shutter speed in Av mode: "auto", 1/60 to 1/250, and 1/250. It is the "auto" setting (default, I think) that can produce slow shutter speeds and therefore "fill" flash. This has caused me problems in the past so I've set the option for 1/60 to 1/250.
Remember that to get sync speeds faster than 1/250, the 580EX II must be set for High Speed Sync.
The G12 (and G1X) are not nearly as sophisicated as a Canon DSLR with respect to flash. This is especially true in Manual exposure. The 580EX II on a DSLR in Manual exposure attempts to automatically fill in and bring up the exposure to the proper amount. The 580EX II on the G12 in Manual exposure simply becomes a manual flash and must be adjusted manually. I suspect that some of this difference also applies to Av mode in the G12.
With the DSLR in active situations, my practice is to set an appropriate and approximate ambient exposure in Manual exposure mode and then fire away with the 580EX II in ETTL mode, tweaking flash compensation as I go. With the G12 (and G1X, I'm sure), my practice is to use Program mode (and face recognition for people) with the 580EX II, again tweaking with flash compensation. For static or studio type shots with the G12 or 7D and flash I use manual exposure and manual flash all the way.
Never as simple as we'd like, is it?