The traditional Mexican drink made from rice, cinnamon, and vanilla is horchata, specifically horchata de arroz.
Mexican horchata is an agua fresca: rice is soaked and ground or blended with water, then strained and sweetened. Cinnamon supplies its recognizable warm aroma, and vanilla is a frequent addition. It is usually served cold and is a classic companion to tacos, enchiladas, and other spicy foods.
The name horchata covers several different regional drinks, which causes frequent confusion. In Valencia, Spain, horchata de chufa is made from soaked tiger nuts rather than rice. Across parts of Central America, horchata may also contain ground morro or jícara seeds, sometimes combined with rice and spices. The Mexican rice-based version is the one intended by this question.
Horchata is not the same as lassi or chai: lassi is a yogurt-based drink, while chai is spiced tea. “Agua fresca” is a broad category that includes horchata, not a competing recipe name for this particular rice drink.