The definition is correct, though not instantly obvious if you see it for the first time. Let me put it this way: a policy is an agent’s strategy.

For example, imagine a world where a robot moves across the room and the task is to get to the target point (x, y), where it gets a reward. Here:

Obviously, some policies are better than others, and there are multiple ways to assess them, namely state-value function and action-value function. The goal of RL is to learn the best policy. Now the definition should make more sense (note that in the context time is better understood as a state):

A policy defines the learning agent’s way of behaving at a given time.

Formally

More formally, we should first define Markov Decision Process (MDP) as a tuple (S, A, P, R, y), where:

• S is a finite set of states
• A is a finite set of actions
• P is a state transition probability matrix (probability of ending up in a state for each current state and each action)
• R is a reward function, given a state and an action
• y is a discount factor, between 0 and 1

Then, a policy π is a probability distribution over actions given states. That is the likelihood of every action when an agent is in a particular state (of course, I’m skipping a lot of details here). This definition corresponds to the second part of your definition.

I highly recommend David Silver’s RL course available on YouTube. The first two lectures focus particularly on MDPs and policies.