Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish with cooking spray.

Beat cake mix, 1 cup water, eggs, oil, lemon zest, and lemon juice in a large bowl with an electric mixer on low speed until moistened. Increase speed to medium and beat for 2 minutes. Pour batter into the prepared baking dish.

Bake in the preheated oven until a toothpick inserted into the center of cake comes out clean, 21 to 26 minutes. Use a fork or chopstick to poke holes in cake; set aside.

Pour remaining 1 cup water into a small saucepan; bring to a boil. Stir in strawberry gelatin mix until dissolved. Mix in 1/4 cup cold water. Pour over cake. Refrigerate soaked cake until completely cool, at least 1 hour.

Make topping: Whisk together milk and lemon pudding mix in a bowl until slightly thickened. Fold in whipped topping.

Frost cooled cake with topping; store in the refrigerator. Top with whole or sliced strawberries when ready to serve.