Can nerve injury mean the same as nerve damage or spinal cord damage?

No: The nerves are outside the axial skeleton, and the spinal cord is contained in the bony structure.

Nerve vs spinal cord: Think of our spinal cord like a tree trunk with many branches coming off ( analogy to nerves). When we damage or injure a nerve it goes to one specific location like a place, on the arm or leg.and we lose the function ( sensation or motion ) to that specific area, when we damage the spinal cord ( just like a tree trunk) we lose the functioning of many nerves coming off that trunk.