Dirt jumping can be done on any little and solidly made bike. Some are almost big wheeled BMXs – one gear, no suspension. You can get a dirt jump bike pretty cheaply. The bike you mention in your post fits into that sort of category. The head angle (the angle of the forks) is can be pretty steep to give quick steering, particularly on street bikes.
Downhill bikes tend to be far more expensive and complicated – long travel forks which will set you back > £200 on their own, ideally a full suspension frame. Probably looking at a grand easily. You’re looking at very shallow head angles on DH bikes, giving nasty steering on the flat, but sorted for DH.
A 4X style bike is somewhere between the two in terms of design and price, and will do a bit of both. They’re normally a little frame like a dirt jump bike, a head angle somewhere between the two extremes, but with a rear derailleur giving you 8/9/10 gears, and a short travel suspension fork.