Make the Lemon Filling: in a medium saucepan, combine the sugar, cornstarch, and salt. Slowly whisk in the water until the mixture is fairly smooth. Place the pot over medium heat and bring to a simmer whisking constantly. Simmer for another minute or two until it has thickened, then remove from the heat.
Whisk the egg yolks in a medium bowl. Slowly pour about 1/2 cup of the hot sugar mixture into the egg yolk mixture, whisking constantly. Then, also whisking constantly, slowly pour the yolk mixture back into the pot with the hot sugar mixture. Return the pan to medium heat and whisk in the 1/3 cup lemon juice, lemon zest, and butter. Return to a simmer, and simmer again for another 2 to 3 minutes until thick and smooth.
Cool the filling for 5 minutes, then pour the filling into the pre-baked pie crust.
Make the Meringue: Place the egg whites, 1 teaspoon lemon juice, cream of tartar, and salt into a large bowl, preferably the bowl of a standing mixer, and beat with an electric mixture on medium speed until soft peaks form. Gradually sprinkle in the 1/2 cup sugar beating all the while until incorporated, and beat just until stiff glossy peaks form. Do not overbeat.
Spoon the meringue over the surface of the pie making sure the entire filling is covered and that the topping touches the inner edges of the crust. Use the back of a spoon or a spatula (offset spatulas are great for this) to make nice swoopy waves in the meringue with some nice peaks.
Bake until the meringue is lightly browned on the top edges of the waves, about 15 minutes. Place the pie on a wire rack to cool, and when it gets to room temperature transfer it to the fridge to chill for at least 6 hours before serving.