Also, I've reevaluated "Greek style" again by practicing a bit and experimenting a little.
First and foremost, I still keep my opinion on the modern style of holding it over the head. I have no idea why, but that style is still relatively ineffective for me compared to the other two. The styles I like better are the depictions shown in the art. I'm not sure if this one does. The closest example I've seen is the Makron cup, which if you look at the hand in relation to the head it looks like the throwing arm side style (I have to think of a better name for it).
Naming ideas: "Greek conventional" (sling held on the throwing arm side) and "Greek unconventional" (sling held on the opposite of the throwing side). The "unconventional" doesn't mean it was unconventional historically, if even used at all. Just unconventional as far as modern use is concerned.
First of which I'll cover is holding the sling on the throwing arm side. The example I found was the Etruscan urn (
http://slinging.org/forum/YaBB.pl?num=1577297750/9 ). It is also depicted with Heracles slinging at the stymphalian birds (and also probably the Makron cup). This is probably the intended "Greek overhand" that we've tried to replicate (though we can't know if it was only one motion). While the Heracles and Makron cup depictions show more traditional hand positions (in being held back at the distance used today), I tried with arm way back like on the Etruscan urn (which might have only been doing that due to length of the sling). Other than that, standard Greek overhand stuff. Remember to twist your body when throwing, that makes a big difference. Don't leave the stone with only gravity acting on it, that causes the stone to fall out. Overall, I found this style a bit more controllable than the other one, but also a bit less powerful. It has still been redeemed, as it is a bit more comparable with other styles, though I personally am not yet much of a fan.
Second is holding the sling on the opposite side of the throwing arm. This is represented on coins from Aspendos and the Kapra slinging relief. Your slinging arm needs to be held behind your head in order for the sling to be straight. It's and awkward position. Other than that, similar advice. Don't let the gravity work too, and twist your body. However, this one is a faster and more violent movement, and I've found it to be more powerful than the other style. It's a bit harder to control, but still perfectly doable. Unlike with the other style, I felt comfortable in throwing larger stones with it. Personally, I like it more than the previous style, though it makes you hold a weird position.
For the "Greek" style as a whole, it's been redeemed but still has some annoying quirks that will keep it from being one I really love. If I wasn't paying attention, I found that was liable to let the stone fall out. This isn't a problem I really have with other styles. Heavier stones were also harder to get that initial momentum to keep them stable. I also forgot to twist my body quite a bit. It sounds stupid, but other styles it's kind of just built in to muscle memory. I felt here that I needed to make more of a conscious effort to remember to do that. One advantage of Greek style is that you can get off shots fast, but I'm not skilled enough to do that with any accuracy or effectiveness. Trying to do it fast makes me accidentally drop the stone a bit more.