I think you're looking at it wrong.

What Atkins is trying to say is that a low carb diet of 2000 calories will burn more fat than a low fat diet of 2000 calories. This is due to the fact that on a high fat diet, not only are you burning calories, but those calories are coming from fat so you're always in fat burning mode. On a low fat diet, your body has to burn up the carbs first before it can begin to start converting your fat to energy.
This is why some people would gain weight on a 2000 calorie low fat diet (though not many because 2000 calories is not too much.

The next statement would be true for ANY diet.
If you consume less that 2000 calories then obviously you're going to lose weight faster because you've got less calories to burn.
But it is easier to eat less than 2000 calories a day on atkins because you do not suffer the same blood sugar level drops as a low fat diet and so do not get hungry as quickly or as often.

Now atkins doesn't really say calories don't matter, what he says is not to worry about your calorie count during INDUCTION.
This is because you must concentrate on getting into this WOE, getting your body into ketosis and learning more about yourself on this diet. Counting calories at this stage just confuses things because you're supposed to eat if you are hungry and eat until you are satisfied. You can't do that comfortably when you keep thinking about how many calories you're stuffing into your mouth.
Eventually your appetite will diminish - especially once you go into ketosis - and you will naturally start to eat less.
I bet by the end of 2 weeks of induction most of us are on or under 2000 calories a day automatically.

Once the two weeks are over, whether you decide to stay on induction or move to OWl, you can start to concentrate on your calorie count a little more because now you'll have a good idea of how metabolically resistant you are and how hnugry you become and how much food you will need to stay satisfied.
If you find you need a lot of food still then stay on 2000 calories a day - you will still lose.
If you find you can eat less, then do, you'll just lose a little bit faster.