At price P1, a quantity of Q1 is demanded. At the lower price of P2, a larger quantity of Q2 is demanded. This is an expansion of demand. At the higher price of P3, a lower quantity of Q3 is demanded. This is a contraction of demand. Only changes in price will cause these movements along the demand curve
A shift from D1 to D2 is an inward shift in demand, so a lower quantity of goods is demanded at the market price of P1. A shift from D1 to D3 is an outward shift in demand. More goods are demanded at the market price of P1
The benefit of the subsidy can go to both the producer, in the form of increased revenue (C-P1), or to the consumer, in the form of lower prices (P1-P2)