A phone that can run apps doesn't count as a dumbphone on my book. Do I use the term differently than most people? When I think of dumb phone, I think of the StarTAC, probably because it was my first. The RAZR was dumb, too, in my mind. A modern example for me is the Punkt MP02. It runs Android, technically, but you wouldn't know it, because you can dobmuch with it but, you know, use it as a phone. That's dumb to me, a phone that is a mostly just a phone.
When the original iPhone came out and didn’t support apps I (unsuccessfully) argued that that meant it wasn’t a smartphone, and it was even less smart than most dumb phones on the market.
At the time I used Sony Ericsson dumbphones, and they had very rich J2ME support - you could set Java apps as a live wallpaper (which iPhone still doesn't allow), multitask (which took years to get to the iPhone) and all this on very very modest (single-digit MB RAM) dumbphone hardware.
I'd personally define "dumb phone" as one that doesn't use the internet at all, period.