There's a difference between Horchata and Agua De Arroz. Agua De Arroz you clean the rice by adding water and draining it a couple of times. You then fill up the pot with one cup rice and 3 to 4 cups water and you add powdered or a stick of cinnamon to it (if you like, but not needed) & a pinch of salt. You heat and every 15 min you keep adding a bit of hot water if needed until the rice becomes soft and you have a lot more water than rice. You then drain or serve with rice and add sugar to taste. It's what mom used to give us when we had stomach aches or food poisoning and couldn't hold any food in. It has a lot of nutrients and at same time it's really refreshing and if you leave rice in it reminds me of a lighter version of arroz con leche. Which by the way, you can also add milk to the finished product to change it up a bit.
I think what you are trying to make is "Horchata" like dksbook said. I'm mexican born and raised and I also add milk and sometimes a little bit of vanilla. You have the right idea =)
This is actually a good recipe - the texture is there, but the flavor needed something. I added some vanilla, and I think it probably would have done well with some almonds. I'll definitely make it again, though, as it's so incredibly easy and workable. Thanks! :)
Here in South Texas, we call this "Horchata" - the "H" is silent. We sometimes make it with milk, too, and often freeze it into popsicles. Some of us like small bits of the rice to be noticeable, and sometimes we just stir powdered cinnamon into the finished drink. And sometimes, we add a bit of rum, which goes great with a strong cinnamon flavor!