I hold it like you would a dagger (!) with the point downwards. You can then stab at the strings "below the bend" to move them into a straight position. But some people over do that motion with too much vigour.
I use a Babolat setting off awl which feels great in the hand. I cant think of using anything other than this type of awl as usually they're not that expensive. If I didn't have one, I would use my fingers like you do.
Head IG Prestige MP; Wimbledon stringing team since '04 - Head Wimbledon stringer '14; Babolat stringing team '10-'14. Twitter - @therqtstringer