Not to mention that displacement of matter from specific areas can cause more "damage" than others. For instance, removing a gram of brain matter would do a lot more damage than removing a gram of muscle fiber.
On a machine, removing a gram of metal from the piston of an internal combustion engine would matter a lot more than removing a gram of armor.

All I am asking is this,

Which formula is a closer representative of "damage"? (damage being defined as displaced matter)

matter displacement = Change in 1/2 * mass * velocity^2 * constant
- or -
matter displacement = Change in mass * velocity * constant

As to everyone saying I should take into account material properties, those are merely constants in the equations that do not change the fundamental equation itself.

But just to make sure there are no more complaints:
Let impacting object (the rod) be perfectly stiff and non-deforming (basically delivers all energy in a single moment in time)... (people are going to question more about this...)
Let impacted object (the wall) be perfectly non-bending/flexible and have it break apart/be displaced in an ideal way (aka wont break apart in fragments, think of it like non-reforming clay).
Let impacted object be unyielding in that even though a dent will be made, the object as a whole is sufficiently well bolted down/kept in place that it will not be moved backwards, thus dispersing energy.
Let impacting object hit the wall in a perfectly dead on way (aka its not going to twist and turn upon impact).
Let impacting object have a arbitrary but set contact area with impacted object. (actual contact area does not matter, since it would only modify a constant in the equation)
Let real life mechanical functions have no influence over the results (its a wall)
Let neurological pain and biological functions have no influence over the results.
Let thermodynamics have no influence over the results
Let aerodynamics have no influence over the results
Let chemical reactions have no influence over the results
Let gravity have no influence over the results
Let magnetism have no influence over the results
Let the alignments of planetary bodies have no influence over the results
Let werewolves and vampires have no influence over the results
Let Navajo sign language have no influence over the results
Let poorly grammatically stated bullet points have no influence over the results
Let forum logic have no influence over the results (wait a sec... cross this one out)

Maybe everything would have been easier if I just said I was looking for a way to realistically calculate impact damage for a computer simulation.

You seem to be making this too complicated then, I'm sure matter displacement = mass x distance will suffice, seeing as it is a literal translation of "matter displacement".