Ref the old laws.
I belive they were taken off the books a while ago. They missed the trial by combat one, but all the bow ones went, I think.
Poaching is a weird one. As far as I am aware, until about 200 hundred years ago the only land you could poach on was Royal land. It was the argicutral revoltion that changed all that, and the current view of poaching started.
As for public owned land, think the problem would be the public bit. It is actual a good question, because what about common land? Which you are allowed to hunt on, if you are a commoner, but think the public bit would get in your way again.
How you hunt is another problem? Hunting with dogs is not illegal. Killing with dogs is. Your dog killing the odd bunny will not get you in trouble but do it a lot, and you could be in trouble. It is one of the reasons the current law is unworkable. If you use a firearm of any description, you have all sorts of laws on carrying them etc to guard for.
The only practical way to look at the law, is that unless you are performing some form of pest control for the land owner, it is illegal. So I would say you are right, hunting in the UK is banned. What is allowed is pest control, which is a legal requirement for land owners. Also looking at some of the information provided, to my knowledge, the only things you are allowed to shoot, are rabbits, minks, grey squirrels, rats, and the pigeon family. The DEFRA information is about a general hunting licease, which seems to be area specfic, and for pest control again.