If you are making a promise not to repeat something, only the second sentence works. You have to use the 'will' form here. 'I won't..' is a promise, an agreement or a resolution not to do something. And seeing you are so fond of formal British English, I can tell you you could even say 'I shan't do it again' with the same meaning. You can't use the present simple in this context, I'm afraid. We only use the present simple with future meaning for fixed arrangements and scheduled events, such as 'My contract ends next March'.

We use about six different grammatical patterns to describe future events (this is even before we get into patterns like "will be doing" or "will have done"), and they need to be understood separately. So no, we can't assume they are interchangeable or applicable to every future event. We use present simple to indicate scheduled events, and it's difficult to imagine a situation that would make "I don't do it again" a future scheduled event. The first sentence states a fact. The second sentence is a promise or prediction.

Only the second sentence is correct.