Say you have a car, when you make a turn the outer radius is going to be a longer distance than the inner radius. That means that the outer wheels are going to need to make more rotations than the inner wheels to make a turn. If you have a solid axle connecting them, then by necessity either the inner wheels are going to slip or the outer wheels are going to skid in a turn.
This is why we have differentials.
Differentials do many things,for instance they act as a reduction gear because your wheels don't spin as fast as your engine shaft (even coming out of the transmission), but the most important thing is that they allow your inner and outer wheels to spin at different speeds. I won't go into the details of how they work, but the key thing to know is the the engine is applying power by revolving the green gear (the purple is not directly connected with either the red or yellow shafts) and if there is unequal resistance coming from the shafts (e.g. in a turn), the green gear will also spin so the power goes the path of least resistance. There are more complex coupling mechanisms (like Torsen differentials which are super cool and I have no idea how they work really), but 99% of vehicles will be using these cheap, reliable and proven mechanisms here.
Let's talk about 4x4 vs AWD then. Apparently there actually isn't a straight forward convention with this terminology as it varies from manufacturer vs manufacturer, but this is how I use it consistently. 4x4 means that your entire powertrain is locked together. You see, because differentials transfer power to the point of least resistance. When turning, this is would be the outer wheels because they can spin more without slipping, but what if one wheel is on ice? Or if it's in the air? All the power is going to be wasted spinning that useless wheel while the differential ensures that no rotational energy is being imparted to the grounded wheel at all. So you lock the differentials and ensure that everything spins at the same time and all of that torque is going into resisting the ground. This does mean that your wheels will slip in a turn, but if you're climbing sand dunes in a Ford Raptor, the surfaces are slippery anyways so this is not a big factor.
You want AWD in a sports car because the main limiting factor in your acceleration isn't how much torque your engine is putting out, it's the amount of grip you have in your tires. So, power to 4 wheels equals 4 times the grip equals better acceleration. Now your R8 is expected to be driven on groomed tarmac, where all the wheels are always mated to a consistent surface, and you don't want to lose traction because your wheels are slipping because everything is locked down in a turn. That's why everything is on differentials and that's AWD.
Now say your Evo lifts off the ground. I believe that it uses a viscous couple mechanism, which submerges the mechanical connections between shafts in a shear-thickening fluid. So, corn starch and water right, if you move slowly through it, it flows around you but if you try to hit it, it acts as a solid. Same idea: if your shaft frequencies are too far apart the fluid solidifies, locking the shafts together. Maybe they use something else fancier these days, but the point is that if your Evo hits a bump, it doesn't shift torque by disconnecting the lifted wheels, it actually locks down everything so that you're not bleeding power through the diffs to a bunch of tires in the air: your AWD Lancer temporarily becomes a 4x4.
So when Richard Hammond breaks a diff in his Toyota, that's easy to understand. He just opens the diff box, severs the powertrain connections, presumably his front and rear are locked so he's not losing any power to a free-spinning shaft and his 4x4 becomes a 2x2. Simple. But when James May loses power to a single wheel, that is mysterious, because you look at that picture of a differential and tell me what parts need to have broken in what fashion in order to enable such a consequence. Any combination I can think of just seems absurd.