"increase in supply"
means a shift of the entire supply curve
an "increase in the amount supplied" is a shift along the supply curve...i think.
Yes you are right to point out that you can have movements along the supply curve as well as shifts of the supply curve, but it is always a game of semantics over which one means which. I mean if you ask me there is no difference between an incease in supply and an increase in the amount supplied. And dont for a second listen to any teacher who says one means this and the other means that. Other than for the purposes of the exam that they have written they are full of shit.
You need to determine the difference based on the question itself. It is usually quite obvious what is meant. An increase in the
amount supplied can come from both a shift in the supply curve and a movement along the supply curve so theres no real distinction than CAN be made. Exams should use terminology such as a '"shift in the supply curve" or "movement along the supply curve" and not simply "increase" or "decrease".