The forward seat is how modern English riders ride (unless doing dressage, but even that is more "forward" than other sorts of riding). The leg is positioned further back and when jumping or moving at speed, the rider moves into the "two point" position, crouched over the horses neck or shoulders. This position allows maximum movement for the horse's spine and makes it easier for the horse to gallop or jump.
A lot of people think this is how English people have always ridden. Not true. The forward seat was invented by Captain Federico Caprilli, an Italian cavalry officer. He analyzed how horses jumped and discovered that the typical jumping seat in which the rider leaned backward to encourage the horse to land hind end first was actually painful and uncomfortable for the horse. His students demonstrated the forward seat at the 1906 Olympic Games and their success...and apparently some demonstrations of bridleless jumping...caused the style to be adopted so widely that almost nobody still rides English in the old "chair seat" style. (It still exists in a few corners, and I'll talk more about that next week when I discuss saddle seat and variations).
So, the modern English style of riding is only a bit over a hundred years old.
The modern jumping position is being demonstrated by a rider in the stadium jumping phase of a three-day event. The rider is looking to the left, presumably because the pair will be turning soon after the landing. The armband contains her medical details, a requirement in three-day eventing (but not seen in pure stadium jumping). Image source: Ronald C. Yochum Jr., via Wikimedia Commons.