In the book of Barto and Sutton, there are 3 methods presented that solve an RL problem: DP, Monte Carlo, and TD. But which category does policy gradient methods (or actor-only methods) classify in? Should I classify them as the 4th method of solving a reinforcement learning problem?
DP, Monte Carlo, and TD are methods of estimating returns. Policy gradient describes methods of learning a policy. So policy gradients serve a different purpose than the other things you mentioned. For clarity, you can use Monte Carlo or TD methods to estimate returns to construct the loss that you get your policy gradient from.