What is lumbar facet arthropathy?

Facet Arthropathy. Could be facet arthropathy/arthritis. There are joints in the spine where the bones articulate and can develop bone spurs and cause the clicking with pain typically with extension. I would consider seeing a pain/spine specialist to see if facet injections or radio frequency ablation would be helpful for you.
Joint degeneration. There is a pair on joints, one right, one left, to the side of the spinal canal (the channel for the nerves to and from the legs). They are behind the disk and vertebral bodies (biggest part of spinal bone). Arthropathy is degeneration, or wear & tear. It is sometimes called arthritis, but arthrosis is more accurate. It can be painful; it can also cause nerve pressure, that is leg pain.
Arthritis of joint. These joints occur ib the spine and bear load and can wear out over time and they can even be injured leading to degeneration which may lead to inflammation and pain in the back.