Grid 1 - State Definition

Define $dp[i][j]$ = number of paths from $(1,1)$ to cell $(i, j)$. This is clean and intuitive. Each cell stores how many ways you can reach it. The answer is $dp[H][W]$.

Notice: you don't need to track which path you took. You only care about how many paths reach each cell. The state $(i, j)$ is enough because movement is restricted to right and down. No extra information is needed to decide the transitions.