I always thought christian healers claimed their powers came from God...blaming Satan would be calling Jesus a satanist, wouldn't it? After all, he healed too, assumedly through powers granted by God...

Angelic healing...I know a few that practices this. My personal, unreligious, extremely biased opinion here is basically people with natural powers, and a strong religious faith, wishing to explain these powers in a way that doesn't conflict with their beliefs.