Technically speaking, if a car has 4 wheels and is an AWD, then it's also 4WD. AWD is just a marketing term that someone came up with when they didn't want their everyday passenger car to be confused with a Cruiser/Paj/Patrol type car.
Generally, people consider off road type cars with things like high/low range gearboxes and lockable front hubs to be 4WD, and cars like your WRX/Liberty to be AWD. I'd put the Stag in the AWD camp, because it doesn't have a high/low range box, doesn't have lockable front hubs, doesn't have good ground clearance and wouldn't be much good off road. Landcruiser is a weird one because it's always in 4WD mode, but I'd still class it as 4WD because of the ground clearance, high/low range box and lockable front hubs. Foresters are a weird one too because some (all?) of them have a high/low range box, but I'd still go with AWD because of the lack of lockable front hubs, average ground clearance and of course it's a Subaru.
Basically, the way I see it, if it's the kind of car that's great for 4WD trails, it's a 4WD. If it's a passenger car that has all wheels driven for performance/safety reasons and it's shite off-road, it's an AWD.