The “Summer Special” car parks (which are available at both North and South terminals) involve you handing over your keys. The ordinary long term car parks are those where you find your own space and keep your keys. Both, as far as I can recall, have no height restrictions. The prices for each of them are virtually identical, and are not much more expensive than the off-site car parks. Both involve a short bus ride to the passenger terminal.
I can understand your reluctance to hand over your keys. I was the same until I tried it. However, I have used the “Summer Special” car parks for about the last three years and always found them to be OK. It is easy as you just leave your keys at reception and your car is brought out ready for your collection when you return.