Preheat the oven to 350 degrees F (175 degrees C). Line three 9-inch round cake pans with parchment paper.
Beat cake mix, water, eggs, and oil in a bowl with an electric mixer on medium speed for 2 minutes. Pour 1/3 of the batter into one of the prepared pans. Mix 1/2 of the strawberry-flavored gelatin and 2 teaspoons strawberry extract into the remaining batter. Divide between the remaining 2 pans.
Bake cakes in the preheated oven until a toothpick inserted into the center comes out clean, 15 to 20 minutes. Line a large wire rack with parchment paper. Invert cakes onto the rack; let cool completely.
Set aside 9 to 12 whole strawberries for decoration. Chop remaining strawberries and pat dry with a paper towel.
Beat cream cheese with the remaining strawberry-flavored gelatin. Add remaining 2 teaspoons strawberry extract and confectioners' sugar; beat until filling is smooth. Fold in chopped strawberries.
Place 1 pink layer on a cake plate. Cover with 1/2 of the strawberry filling. Add the plain cake layer. Cover with remaining strawberry filling and position second pink layer on top.
Make frosting by beating heavy cream, confectioners' sugar, and vanilla extract together until dense enough to spread. Spread frosting over the top and sides of the cake. Place a piece of parchment paper on top of the cake. Run spatula over the paper to smooth the top. Peel off the paper.
Pipe 9 to 12 swirls of frosting around the rim of the cake; place whole strawberries on top.
