People use the two names as if they mean the same cup. They do not.
Cutting chai
Cutting means a half portion. The word is about quantity: a small glass, strong enough that half is plenty.
Kadak chai
Kadak is about strength. The leaves boil longer with the milk, so the colour goes deep and the taste turns bold. Ginger and cardamom go in early so they cook with the leaves.
Cutting tells you how much. Kadak tells you how strong.
At the cafe every chai is kadak by default. Ask for it light and we will pull it off the flame sooner.