Is nerve injury the same as nerve damage or spinal cord damage?

Possibly: Nerve injury implies some kind of damage to the nerve itself. While spinal cord and nerves are both nervous tissue, nerves typically are structures that have branched outside the spinal cord. Injury to the sciatic nerve, for example, is not considered a spinal cord injury. They are very different structures.

Nerve vs Cord injury: No, nerve injuries refer to nerves located outside of the spinal canal and have a significant chance for regeneration. The spinal cord does not have the same potential for repair.