My reproduction sling is made from eight strands of twisted cotton with a leather finger loop and a tear drop shaped release tab. I have modified the pouch a little by adding a bridge at the center of the split pouch design to keep the stone from accidentally releasing out of the bottom of the pouch during the wind up.
I drew the design out on heavy paper and have included it so that it can be copied. The release tab was made from a piece of material cut out from the center of the pouch.
EDIT: With a little feedback and research this style of sling is from the Tunguska / Siberian region of Russia.