I tried searching through a journal database to see if AC is contagious by physical contact. Unfortunately, I couldn't find anything that confirmed or refuted this. The only Google results I found were ones related to that AC cure site which said it wasn't contagious.
Here's my personal take. Generally, mouth corner cuts are not directly caused by bacteria or a fungus, but by cold, heat, the foods we eat, or something self-mechanical. Usually the cuts heal but if they don't, it leaves the cuts vulnerable to infection and so a fungus colonizes in the cuts and prevents it from closing.
So perhaps what the AC site was really saying that someone with AC can't give someone else mouth wounds by personal contact. (He may pass on the fungus, though, and that may or may not be harmful.)