Answer: You should report the joint injection with only 20610 (Arthrocentesis, aspiration and/or injection; major joint or bursa ...). Because 20610's description states "Arthrocentesis, aspiration and/or injection," the code includes the injection.
In contrast, 90782 (Therapeutic, prophylactic or diagnostic injection [specify material injected]; subcutaneous or intramuscular) describes a therapeutic, prophylactic or diagnostic injection. You would use 90782 for a preventive shot, such as Synagis for respiratory syncytial virus, or a restorative treatment, like Decadron for croup.
Don't forget to bill Medicare carriers with the corresponding G code instead of 90782. In 2005, CMS replaced 90782 with G0351 (Therapeutic or diagnostic injection [specify substance or drug]; subcutaneous or intramuscular).
Remember: You should separately bill for the substance the FP injects into the joint.