What correctly distinguishes qaḍāʾ from kaffārah?

Kitab al-Sawm: Fasting Hours, Invalidators, Qada and Kaffarah

• Qaḍāʾ replaces the fast; kaffārah is additional expiation
• Qaḍāʾ is charity only and kaffārah intention only
• They always mean exactly the same duty
• Kaffārah is due for every valid fast