How do you diagnose facet joint syndrome?

With difficulty. This is a diagnosis doctors disagree about. Mri and ct may show degeneration (aging, arthritis) of the small joints behind the disk and nerve channel. In some people this is painless. In some folks they may be a source of pain. Injection of medication (anesthetic or cortisone like) into the joint or around the small nerve supplying the joint is viewed by some as evidence of facet joint syndrome.
Typically. Pain with lumbar extension and rotation. If it hurts when you lean back and rotate your back then it possible you could have some facet joint pain. I would see a spine/pain specialist to determine what options you may have for treatment.
By exam history test. By getting a good clinical history and doing an exam that notes pain with prolonged standing and sitting and made worse with extension and sometimes twisting followed by imaging including x-ray, MRI & ct and then further procedure testing in terms of facet blocks to see if pain is relieved by this procedure. This syndrome is usually associated with other back issues.