I've been warming to the idea as a useful insight, but I'm still pretty confused; it feels like there's a useful (but possibly quantitative and not qualitative) difference between myself (obviously 'conscious' for any coherent extrapolated meaning of the term) and my computer (obviously not conscious (to any significant extent?)), which is not accounted for by saying merely that consciousness is the feeling of information processing.

Why do you think your computer is not conscious? It probably has more of a conscious experience than, say, a flatworm or sea urchin. (As byrnema notes, conscious does not necessarily imply self-aware here.)