My teacher told me that drinking water that is room temperature is better for you then drinking cold water, but she did not tell me why that is. I want to know because I often drink cold water (with ice), because I can't stand water that is room temperature or warm.
Your teacher was passing on a myth.
This false belief might be related to the equally false belief that hot exerted horses should not be fed cold water. It could also be down to someone downing a big glass of ice-cold water in one go as a child, get a brief stomach ache and spin a health-advice on it.