Currently I've settled on simply using both. I use the diagonal overhand cast with rotor with a shorter and slightly thicker sling of an arms length long for target practice and accuracy, with a preference for ball shaped stones or spherical clay shot. I use the tight grip as it is very secure with rotations. My accuracy this way feels the best as it follows a very instinctive throwing motion. But using this sling and this projectile feels subpar for distance

But for distance I will use a different sling and style, I will use a powerful full body Greek cast with a very thin sling and a very wide grip, with the sling being the length of my extended arm to almost the opposite shoulder. The lack of rotations means I can get very good pouch angles with the extrem thin slings, using a sidearm release with a straight arm (first the hip, followed by the torso, and then primarily the shoulder, then followed by the arm.)  The distance I achieve is greater than anything I have reached ( around 220 meters so far with 60-80 gram elongated pebbles on flat ground,with sea level altidue and a side wind.) The general ammo I would use for this sling is sub 100 gram elongated pebbles, or lead and clay biconicals or tear drops.
But using this sling and this way of throwing feels sub par for short distance target practice as it is not the way we throw naturally, but the distance I get is doubled compared to my target sling.

Some wild rabbits are really stupid, other kinds of rabbit or hare will sprint away at the seeing just slight motion from a hundred meters

Last night I was slinging at a target with a simple 1-2-3 rotor and then diagonal overhand cast (3 rotors in 3 counts then cast on the third count). I think this is the way for the rotor, instill in yourself a select certain amount of turns and stick with it instead of spinning it around a number of times randomly. The counting to 3 and then throw on the third might also help with clearing the mind from distractions

I like this demonstration of Greek style.. I do similar
https://youtu.be/e12pRrHitJ0
I have to coil my body backwards for a distance cast.

The hares around where we used to live were very acclimated to humans and even dogs. You could get within 3m and they wouldn't move. I even spun an empty sling above my head and they did mostly run off, but it would have been very possible to hit one. Birds as well I have seen be completely fearless.

   Apache gives you a very quick release that doesn't give game a chance to react, of course.  And it's reasonably accurate. Its drawback is range.  I can get twice as much distance with other styles.

I use both Greek without spin and Byzantine and Balearic style

yeah it's sort of horses for courses.
I do windups for distance slinging, but not for most things.

You don't really gain anything with a short to medium sling.

But with a long distance sling the gradual speed buildup does really help.