How do you apply a pure torque? A pure torque does not have a point of application. For example, in the beam shown below, the reactions are same regardless of the torque placement:

But how do you apply torque in reality?
For example if you use a screwdriver or a double force the beam deformation is different based on where you apply the torque as shown below:

Another example, how do you calculate the reaction forces in the systems shown below:

So, does pure torque exist in reality?
 A: 
How do you apply a pure torque

A pure torque or pure moment, is referred to as a force couple, or simply a couple. It is achieved by applying two equal and opposite parallel forces of magnitude $F$ separated by a distance $d$. This produces a net torque of $Fd$ with no net force. By convention, the symbol for a force couple in statics, $M_c$ in the diagram, simply shows the moment produced by the forces and not the forces themselves.
That said, an ideal (frictionless) pin or roller support offers no torque reaction, i.e., the sum of the moments about the pin or roller is always zero. So be aware that the torque you show at the pin location in the bottom diagrams is within the beam to the left of the pin, not at the actual pin itself.  The vertical reaction force acting up at the pin opposes one of the two equal and opposite parallel forces that produce the couple. The other would be applied further to the left of the beam. What you show as torque at the pin would be free rotation of the beam in response to its distortion. See the diagram below.
Finally, it is not true that the pure torque does not have a point of application. It's just that the point of application is irrelevant when determining the requirements for static equilibrium.  For that reason, a couple is referred to as a "free moment vector", as I explained in my answer to you previous post. On the other hand, the point of application must be taken into consideration when determining bending moments and stresses and their locations in deformable solids.
Hope this helps.

A: 
A pure torque does not have a point of application.

This is incorrect (as you show in your second example). A better statement is
"The location of application of a pure torque doesn't matter in certain analyses, such as when balancing moments around any point of a rigid free object."
The qualifiers are satisfied in your first example. But the beam in your second examples isn't a rigid free object, and the point of application of the torque does happen to matter.
A: I think that the heart of this paradox is that "pure torque" is in idealization (as is "rigid body")  in fact "pure torques" do not exist (anything applying a torque applies finite forces at points that have a finite distance apart).  What we are seeing with the latter example is that the idea of applying torque at a point is useful only to the degree that the rigid body approximation (e.g. r_ij is fixed for all the elements comprising the body) is valid.
If the body can deform, then how it deforms is determined by where the forces act.  You need to work with pressures and stress on the object in order to determine how it deforms (e.g. a hex nut that is stripped will not deform the same way as a phillips screw because the stresses on the object are different)
