With English pads, as long as your saddle fits well (which it should!), the pad is just there to protect the saddle from sweat. The panels (the squishy-ish part of the saddle that sits on the top of the horse's back) are the cushion for the back, that's why they're squishy. :) Basically, if your English saddle fits just right or is slightly too narrow, a thick pad is going to make that fit even worse. If the saddle is much too wide, a thicker pad might be able to help but if the saddle is really that much wider than it needs to be, you're going to have other fit issues that are unfixable with a pad.
I'm not super familiar with half-pads but I believe that they are used to cushion the spine without adding too much padding in other areas. I'm not 100% though.
For a pad, I'd just get a normal (one of my english pads is this style and I love it - it feels more cushiony to my fingers than normal english pads but it's no thicker than the average AP pad) english pad. http://www.horse.com/item/roma-ecole...pad/SLT900888/
Whatever you get, I'd get more than one since english pads needs to be washed regularly (I wash mine every week so I have 3 pads, two are washed at a time while one is being used, etc).
Western saddles don't have that built in padding, that's why western pads are much thicker than English ones.
I have a Diamond Wool pad that I adore. The only bummer about mine is that it's 1.5 inches thick and Lacey's so wide that any pad over 3/4 inch throws off saddle fit quite a bit. Anyway, I can't use that one right now but I'm holding on to it for when/if I ever get another horse.
You'll find that a lot of people like neoprene pads or other kinds of pads like that which I guess works for them. For me, I find that wool breathes pretty well and horses like it, so why change? Haha I also like how wool conforms to the horse's back over time.
When looking for a pad: look for one that contoured to the horse's back and preferably has wear leathers - I find they help the pad look new for longer.
Here's a link to one like mine: TB25- Diamond Wool Contoured Cowboy Saddle Pad