Thanks. I looked into 'The Google' and found a couple of articles that said the same. I still can not get past the idea of ALL the pictures not having horrid amounts of red-eye.
Well, a lot of ring flash shots are taken fairly close up so that the ring catchlights can be fully appreciated, and you'll usually have 2-3 inches between the lens and the actual bulb, so that helps it to not be too direct. I rarely get redeye while using on-camera flashes aimed directly at the subject because it's a good few inches above, unlike P&S cams where the lens is literally right next to the flash.
I suspect it could also have something to do with the relative size of the light source as well: a ringflash is effectively, what, maybe 8x the diameter of the head of a Speedlight? I don't get redeye with a medium softbox positioned right over the camera either, nor a big octabox right behind the camera. Not sure I've gotten redeye using Speedlight modifiers like Lumiquest ProBounces or mini softboxes either.
Honestly, none of the ringflashes I've used have modeling lights, so I'm not convinced that's a factor. But I'm just guessing here.