Some people do GREAT with a CD spindle. My best friend still prefers her CD spindle over any other! I was a different story - I got really frustrated with drop spindling until I got a different kind of spindle. (Not to put anyone off, I just mention this in case you try a CD spindle and don't get the knack of it.)
I looked at all the resources in the sticky and it seems that high whorl creates thin yarns (spins faster) and low whorl creates thick yarns (spins slower). If you want to make a thick yarn you also need to use a rolag (a rolag uses short hairs that are combed with a lot of loft creating a light, thick yarn) rather than roving (roving uses long hairs and makes the yarn tight, dense and strong). There is also the Navajo spindle spinning.
From what I've read, not what I've tried (which right now is nothing lol)
Hmm, from the sounds of it, a low whorl/whirl (?) spindle with a large diameter, so that you can make thicker yarn. I know from experience that making thinner yarn is a pain in the butt! And I was doing it with a high whirl drop spindle, ...it's definitely like lace weight!