Make the Cake:

1. Preheat the oven:
   Preheat your oven to 350°F (175°C). Grease and flour two 9-inch round cake pans, or line them with parchment paper.

2. Mix dry ingredients:
   In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, and cocoa powder. Set aside.

3. Cream the butter and sugar:
   In a large bowl, beat the softened butter and granulated sugar together until light and fluffy, about 4-5 minutes using a hand or stand mixer.

4. Add the eggs and vanilla:
   Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.

5. Add the dry ingredients:
   Gradually add the dry ingredients to the wet ingredients, alternating with the buttermilk. Start and end with the flour mixture. Mix until just combined.

6. Add the purple color and vinegar:
   Stir in the purple food coloring, vinegar, and lemon juice (if using) until the color is evenly distributed.

7. Bake the cakes:
   Pour the batter evenly into the prepared cake pans. Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean. Allow the cakes to cool in the pans for 10 minutes before transferring them to a wire rack to cool completely.

Make the Cream Cheese Frosting:

1. Beat the cream cheese and butter:
   In a large mixing bowl, beat the softened cream cheese and butter together until smooth and creamy.

2. Add powdered sugar and vanilla:
   Gradually add the powdered sugar, 1 cup at a time, and beat until smooth and fluffy. Stir in the vanilla extract and a pinch of salt.

3. Frost the cake:
   Once the cakes have completely cooled, spread a layer of cream cheese frosting on top of one cake layer. Place the second cake layer on top and frost the top and sides of the entire cake with the remaining frosting.