These terms are actually becoming obsolete with these electric drive units.
Historically, "4x4" was used to refer to systems that would mechanically lock torque bias equally front and rear, while "AWD" was used in vehicles capable of moving only a portion of available torque from one axle to the next; usually through a clutch system.
The Lighting, as with the Tesla that I drive now, is capable of providing 100% torque to all four wheels at once, operationally putting in in the same category as a 4x4 with fully locked differentials.
Since the Lightning (and the model Y) have a "mode" that equalizes torque front to rear, they are "4x4s". It could be argued that since the tesla model 3 has no such "mode" (yet), it is the new definition of "AWD". This said, the lines between the two have become significantly blurred with advancing technology. A better question becomes, "does the vehicle have an Off-road mode that provides for instant torque to all the wheels?... which is all you can ask from the very best 4x4s.
In the case of the Lightning, the answer is yes.
