An increase in demand causes an increase in the equilibrium price

This is a false statement. Both supply curve shifts only when there is a change in non-price determinants of supply. You move along the supply curve when there is a change in the price of the good. The reason is simple. Price is on the vertical axis. Thus, a change in price represents movement along the supply curve. Conversely, non-price determinants such as input prices, technology, are not measured in any of the axis. Hence, you shift the supply curve when one of non-price determinants of supply changes. When supply curve shifts, there is a change in supply, whereas when you move along the curve, there is a change in quantity supplied. Thus, the increase in equilibrium price will then cause an increase in quantity supplied, not in supply.