I've heard that if you rinse the pan (don't dry it) with cold water right before you put the milk in, it's supposed to help keep it from burning.

An extremely heavy-bottomed pan that has a thick carbon steel, cast iron, copper or aluminum core for even heat distribution is a must. If you can hold the pan with one hand, it isn't heavy enough.
Remember, milk is chock full of sugars, minerals, casein, albumin, globulins and a mixture of mixed triglycerides...all these are prone to coagulate or otherwise stick to the inside/bottom of the hot vessel. A commercial dairy mitigates this problem by heating the milk with direct injection of culinary grade steam into the milk. At home you might accomplish this with a nice cappuccino machine. Barring a cappuccino machine a high quality non-stick pan interior helps the stuck-on stuff to easily release. Stirring is not enough; you need to be able to effectively "scrape" the stuck-on milk components off the inside of the vessel continually as you stir! Any inside surface area that does not get scrapped will accumulate a coating that will scorch as you have correctly noted no matter how low or gradual the heat is applied. Try a stiff siliconized heat resistant scrapper with a fine edge. A commercial dairy vat uses motor driven agitator/ scrappers that work like windshield wipers to keep the coating from sticking and scorching. The combination of a very heavy pan, non-stick interior, medium-low heat and effective stir/ scrapping of the bottom and sides of the vessel will mitigate scorching. Also, a double boiler set-up in addition to all the former attributes will help assure gentle and even heat if performed over simmering (not boiling) water.