Rinse the basmati rice under cold running water until the water runs clear to remove excess starch. Combine the rinsed rice with 300 ml water and a pinch of salt in a saucepan, bring to a gentle simmer and cook for about 8–10 minutes until the rice is just tender; drain any excess water and return rice to the pan.
In a separate pot, warm the coconut milk with 400 ml water, the sweetener and vanilla over low heat until just below a simmer. Do not boil vigorously — heat gently so the coconut stays silky.
Add the warm coconut mixture to the partially cooked rice in three additions, allowing the rice to absorb most of each portion before adding the next. Keep the pan on low heat and stir frequently so the rice softens evenly and the mixture thickens to a creamy consistency, about 15–20 minutes total after adding the coconut liquid.
When the final addition is mostly absorbed and the rice is soft but individual grains still hold shape, remove from the heat and fold in the shredded coconut. Taste and adjust sweetness or vanilla, and let the pudding rest briefly so flavors meld.
While the rice finishes, place chopped ripe mango with the lemon or lime juice into a blender and purée until smooth; add a small amount of sweetener only if needed. If mangoes are slightly underripe, warm them with a tablespoon of water for a minute before blending to soften.
Spoon warm pudding into bowls or parfait glasses and top with a generous layer of mango purée. Garnish with extra mango cubes, toasted shredded coconut, or chopped pistachios; serve warm or chill for at least 30 minutes for a colder dessert.