Homemade yogurt & flavours - Keeping it thick
Creating the right environment for bacterial fermentation

Yogurt is made using a few different bacteria which excel at digesting the lactose in milk at warm temperatures. Each of those factors (culture, temperature, composition of milk,) is important in creating an ideal environment for bacterial fermentation. 

You would not try fermenting yogurt using yeast or a bacterial culture different from yogurt starter, and you wouldn't expect the yogurt to ferment if you held it your refrigerator instead of a warm thermos. The composition of the milk is equally important. If you change a major component of the milk, such as by adding sugar, some fermentation may happen but the result will not be what you expect.

"Live active culture" vs. actively culturing

There is a difference between bacteria that is "alive" or "active" and bacteria that is currently doing something. 

Imagine dry baking yeast in a canister. The yeast is alive, but inert. It's not in an environment where it can do anything. Add moisture and carbohydrates for it to act on, and it goes nuts. Put that dough in the refrigerator, and it slows way down. Until you cook the yeast it doesn't "die" or "deactivate" it just has various states of activity depending on its environment.

The yogurt culture is alive when you add it to the milk, alive when you're done incubating it, alive when you add sugar, and alive when you eat it, But it can't turn milk into yogurt unless you give it a very precise environment -- the right culture, the right temperature, the right environment in the milk, including the right amount of sugar.