The O'Tom hook tick twister is very good, I keep one at the yard and one at home (for the dogs). I bought mine at our vets. Cost - less than £4. I expect to take 12 ticks off my horse daily at the moment, mainly around the face.
The suffocating/burning/killing with alcohol methods all have drawbacks - the tick may well vomit its stomach contents into the horse's bloodstream. Pulling the tick out with your bare hands puts you at risk from blood or regurgitated stomach contents - for example, if you have a tiny break in your skin, a cut or graze.
The big risk - much bigger IMHO than the risk to the horse of leaving the head embedded - is that the tick is carrying the bacteria which causes Lyme Disease (Borrelia Burgdorferi). This is a seriously nasty bug, a spirochete (like the bug which causes syphylis!!) In humans it can cause symptoms soon after infection, and then chronic illness for years down the line. It is difficult to diagnose, there are blood tests available but they seem to throw up a lot of false negatives - that is, you can have a negative blood test even though you have the disease.
Lyme disease is a world wide problem, in Europe the strain of Borrelia is different to that in the US, and the long-term problems are more neurological/cardiac, while in the US arthritis is the best known symptom.
If you want to know more, I'd suggest having a look at
http://www.bada-uk.org If you go to 'downloads', you'll find they have a leaflet on Equine Lyme Disease.
I try *never* to get bitten, I basically keep my lower half covered - trousers tucked in/gaiters/chaps/whatever.
Good luck, and stay tick-free!
mary