If you have time, soak the rice in water for a few hours, then drain. This step is optional.
Add the rice and 2 cups of water to a large, heavy pot over medium-high heat. Cook, stirring constantly so the rice doesn't stick to the bottom, about 5 minutes, until the water is absorbed.
Stir in the coconut milk and milk, then add the cinnamon sticks, salt and lemon zest. Reduce the heat to medium and cook for 5 minutes, stirring the whole time.
Add the evaporated milk and sweetened condensed milk. Reduce the heat to low, cover, and let it gently simmer for 15 to 20 minutes, stirring every 5 minutes, until the rice is tender and the pudding is creamy.
Remove the pot from the heat right away, while the pudding still looks a little runny; it thickens as it cools. Remove the cinnamon sticks and stir in the vanilla. Move the pot to a cool burner so it stops cooking.
If it's thicker than you like, stir in a splash of cold milk. Serve warm or chilled, sprinkled with ground cinnamon and topped with fresh fruit.
Notes
Optional: soak the rice in water for a few hours before cooking, then drain. Leftovers keep up to 4 days covered in the fridge; stir in a little cold milk before serving to make them creamy again.