Is numbness the absence of one or all touch sensations?
The definition of numbness has been answered by yourself, and I will focus on the second part of the question.

In terms of the underlying physiological mechanism behind numbness I think it's good to narrow the question down and focus on local anesthetics, which numb a local area of the body for minor surgical procedures. Cocaine and related compounds have been shown to block not only pain, but also warmth, cold and touch perception, as well as blocking motor function. Cocain and related compounds are basically voltage-gated Na+ channel blockers, but also block K+ channels to some extent. Their affinity to various different channels differs, explaining the differential effects. E.g., pain is blocked more effectively than motor functions (Scholz, 2002).

We all know the feeling of numbness when our limbs are asleep due to restriction of blood flow. Likely all neural activity is blocked here too, as the cutting off of oxygen affects all neurons.

Given the broad range of effects of local anesthetics and restriction of blood flow, I would tentatively conclude that numbness can be interpreted as a reduction of sensitivity of all the senses.  

Reference
- Scholz, Br J Anaesth (2002); 89; 52-61