Slings are found on Trebuchets and Onagers. (Onagers are also known as Mangonels, but they are the same thing mechanically.)
The do indeed work the same way as a staff or hand sling. The only difference lies in the speeds, materials used, payload being hurled (cast), method of release, etc.
Hand slings don't have a "starting point" per se, although hanging straight down could be considered the start. Some folk hold the pouch during the start though, so that leaves the question open a bit.
Onagers normally start with the sling hanging staight down. In some cases this presents a problem since the sling is often longer than there is room to hang freely in. This requires either digging a hole or setting the machine up a bit higher in the air.
Trebuchets start with the sling laying in a trough, horizontally. As the throwing arm rotates, the sling is dragged along the trough until it picks up enough speed to lift the pouch upward. (Which usually happens very quickly.) After that, the sling continues to rotate around the end of the throwing arm until it reaches the release point.
The release point is determined by a couple of variables, which I'll discuss if needed, but siffice for now to say that it should be at a point where the sling releases at about 45 degrees. (Chris, the optimum release point is not 45 degrees, but close enough for this discussion.)
So, for an Onager the sling rotation angle is about 135 degrees. For a Trebuchet it is about 225 degrees.
For those interested in Trebuchets, visit my web site for a lot of information.
http://www.ripcord.ws