To prepare the cake, coat 2 (9-inch) round cake pans with cooking spray; line bottoms of pans with wax paper. Coat wax paper with cooking spray.
Place 3/4 cup sliced strawberries in a blender; process until smooth.
Place 2 cups granulated sugar, butter, and gelatin in a large bowl; beat with a mixer at medium speed until blended (about 5 minutes). Add eggs, 1 at a time, beating well after each addition.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and baking powder, stirring well with a whisk. Add flour mixture to sugar mixture alternately with milk, beginning and ending with flour mixture. Stir in pureed strawberries and 1 teaspoon vanilla.
Pour the batter into prepared pans; sharply tap pans once on counter to remove any air bubbles. Bake at 350° for 45 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes on a wire rack, and remove from pans. Peel off wax paper, and cool completely.
To prepare filling, combine whipped topping, 2 tablespoons granulated sugar, and 1/2 teaspoon vanilla in a small bowl.
To prepare the frosting, beat cream cheese, 1/4 cup butter, and 1/4 teaspoon vanilla in a medium bowl with a mixer at low speed just until well-blended (do not overbeat). Gradually add powdered sugar, and beat just until well-blended (do not overbeat).
Place 1 cake layer on a plate; spread with 3/4 cup filling. Arrange 3/4 cup sliced strawberries over the filling; top with remaining cake layer. Spread the remaining filling over top of cake. Spread the frosting over sides of cake. Arrange quartered strawberries on top of cake. Store cake loosely covered in refrigerator.