Will a tail bone injury fall in the category of spinal cord injury?

No. If all that was injured is the coccyx. The spinal cord ends in the upper lumbar spine. The coccyx or "tail bone" is a small boney structure off the end of the sacrum. There are no associated spinal cord components that far down the spinal collumn. Injuries to this bone and surrounding soft tissues can be very painfull however, and perpheral nerves can be involved.
No. An injury to the coccyx (tail bone) is not a spinal cord injury. The spinal cord ends at around the 1st or second lumbar vertebrae all the way up in the lower back.