We study a three-strategy spatial evolutionary prisoner's dilemma game with imitation and logit update rules. Players can follow the always-cooperating, always-defecting or the win-stay-lose-shift (WSLS) strategies and gain their payoff from games with their direct neighbors on a square lattice. The friendliness parameter of the WSLS strategy-characterizing its cooperation probability in the first round-tunes the cyclic component of the game determining whether the game can be characterized by a potential. We measured and calculated the phase diagrams of the system for a wide range of parameters. When the game is a potential game and the logit rule is applied, the theoretically predicted phase diagram agrees very well with the simulation results. Surprisingly, this phase diagram can be accurate even in the nonpotential case if there are only two surviving strategies in the stationary state; this result harmonizes with the fact that all 2×2 games are potential games. For the imitation dynamics, we found that the effects of spatiality combined with the presence of two cooperative strategies are so strong that they suppress even substantial changes in the payoff matrix, thus the phase diagrams are independent of the cyclic component's intensity. At the same time, this type of strategy update mechanism supports the formation of cooperative clusters that results in a cooperative society in a wider parameter range compared to the logit dynamics.