Bring water and salt to a boil in a saucepan over high heat. Add rice. Reduce heat, and simmer, covered, until water is absorbed, about 20 minutes.
Meanwhile, brown pork in a large skillet over medium-high heat, stirring to break up lumps, 5 to 7 minutes. Transfer pork to a plate; drain grease from the skillet.
Heat oil in the skillet over medium-high heat. Stir in onion, bell pepper, carrot, water chestnuts, and ginger; cook, stirring, until onion is translucent, about 5 minutes. Stir in pork, teriyaki, and honey; cook, stirring, until liquid has evaporated, 3 to 5 minutes.
Fluff cooked rice with a fork, then fold in 1/4 cup cilantro, coconut milk, and lime juice.
Top lettuce leaves with pork mixture and rice. Garnish with lime wedges and chopped fresh cilantro.