How do the doctors describe chronic lumbago?

Lumbago, old term. This chronic pain in the distribution as you suggested is the result of an irritated nerve or facet joints or other injury typically in the lumbar spine (low back) which are caused by herniated disks, spinal stenosis or degenerative disc disease, etc requiring further evaluation by a spine specialist and may be candidate for facet injections/radiofrequency ablation and epidural steroid injection.
Old, common term. "lumbago" is an old, non-medical word used to described low back pain. Low back pain can take on many forms and have many causes. Any recurrent and/or chronic low back pain should be evaluated by a physician. If necessary, your doctor can recommend an appropriate specialist.
Low back pain. Lumbago=low back pain some people may also use it to describe mild lumbar disc degeneration.