Annealing has nothing to do with the amount of carbon in a material.
Annealing is a comprehensive term. The process of such a heat treatment may be: to remove stresses; to induce softness; to alter ductility; toughness; electrical magnetic, or other physical properties; to refine the crystalline structure; to remove gases; to produce a definite micro-structure. ( From http://www.metal-mart.com/Dictionary/dictleta.htm)
It can be used to describe heating and cooling processes for metals and non-metals alike and appears to have no direct relationship to the presence or absence of carbon.
In the broadest sense annealing is correctly used when described in laser marking as it heating followed by cooling to produce a definitive micro-structure.
Carbon Migration is borrowed from welding where carbon migration is considered to be a significant factor in determining the life of a weld and appears to be well studied phenomena.
Carbon migration (AIAG Spec B-17) and laser discolorization (SAE AS9132) are standard terms in our industry to describe the effects of darkening material.