The first expresses what someone does for fun. The second expresses a preference for a particular task when the chores are being divided among persons in a household.

Very little difference. Most of the time we would say, "I enjoy cooking." This means that I enjoy preparing the food to eat. "I enjoy doing the cooking" could mean "I enjoy having the responsibility (or job / task) of cooking. Here is another example. I enjoy paying the bills. A person probably doesn't enjoy paying out the money, but they could enjoy the job of paying bills so they know where their money is going, or how it is spent.

This is also what I was tryna figure: I'll talk by then, you just keep your mouth shut. vs I'll do the talking by then, you... shut. "I enjoy cooking" is for sure what people like to say though.