Rotator Cuff Tendinopathy or Tear: The patient's presentation of sudden onset sharp pain at the shoulder joint with an aching sensation radiating down the arm, exacerbated by certain shoulder positions and rapid movements, is highly suggestive of rotator cuff tendinopathy or a tear. The limited active internal rotation and shoulder abduction, along with the inability to move the hand off the back on external rotation against resistance, further support this diagnosis.

Subacromial Bursitis: The patient's pain pattern and exacerbation with rapid movements could also be consistent with subacromial bursitis, which often presents with pain and limited mobility in the shoulder.
Early Stages of Adhesive Capsulitis: Although the patient has a relatively preserved passive range of motion, the painful active range of motion and limited internal rotation could suggest early adhesive capsulitis, especially given the history of prior shoulder issues.

Thoracic Outlet Syndrome: Although less likely, thoracic outlet syndrome could present with shoulder and arm pain, and it is crucial to consider this diagnosis due to its potential for significant morbidity if missed.
Referred Pain from Cardiac or Pulmonary Sources: Given the patient's age and history of smoking, although the patient denies chest pain or shortness of breath, it is essential to consider referred pain from cardiac or pulmonary sources, as these conditions can be life-threatening if not promptly addressed.
Infection or Septic Arthritis: Although the patient does not have fever or significant swelling, infection or septic arthritis must be considered, especially in the context of recent steroid injection, due to the potential for severe consequences if not treated promptly.

Polymyalgia Rheumatica: This condition could present with shoulder pain and stiffness, although it typically involves both shoulders and is accompanied by other systemic symptoms such as fever and weight loss, which are not present in this case.
Neoplastic Processes: Although rare, neoplastic processes affecting the shoulder or surrounding structures could present with pain and limited mobility, and should be considered in the differential diagnosis, especially if other diagnoses are ruled out and the pain persists or worsens over time.

Imaging Studies: Order an MRI of the right shoulder to evaluate for rotator cuff tendinopathy or tear, and to assess for other potential causes of shoulder pain.
Physical Therapy: Refer the patient to physical therapy to improve range of motion and strength, and to address any biomechanical issues contributing to the pain.
Pain Management: Consider prescribing NSAIDs or other analgesics to manage the patient's pain, and discuss the potential benefits and risks of steroid injections.
Lifestyle Modifications: Encourage the patient to avoid activities that exacerbate the pain and to modify their daily activities to reduce stress on the shoulder.
Follow-up: Schedule a follow-up appointment to reassess the patient's symptoms and adjust the management plan as needed.