I also prefer the older method of the Atkins diet as it works better for my lifestyle. Some will tell you that you absolutely, MUST do it the new way or it won't work. Bull.
One example of a difference between old school Atkins and new Atkins: The old way you weren't allowed any coffee for Induction, now, apparently, you can have coffee.
Just do the one that works for you and don't worry about anyone who may be so militant about the whole thing.
The first two weeks you want to stay under 20 grams of carbs a day...(or longer if you want to stay on Induction longer), but you will not eat this way the rest of your life. You will gradually add some carbs back in to your life; it's just not realistic to eat under the Induction plan forever. Basically you're trying to get your body in to Ketosis, which changes the way it handles the blood sugar and what not. I'm not going to get in to it all here and be technical, just get the book; whatever version you want.
Also, there has been a major misconception about the Atkins Diet since the beginning. Never, anywhere, did Atkins say you could eat a pound of bacon a day. That was never, ever, EVER said.
What you want to do is eat low carb veggies, eggs, meat, cheese to get your minimal carbs, fat and protein. You also want to take a vitamin and some Omega-3 capsules. The weight will fly off but do not look at it as a quick fix and then you go back to how you ate before, you want to establish a lifestyle of eating way less carbs than you normally do. Just look at the amount of carbs in ONE serving of pop, (soda), or fruit juices. You can have all your carbs for the day with just one of those.
You want to be cutting out bread, pasta, sugar, rice, fruits, (for Induction at least), starches, (no potatoes and corn on Induction for sure)...pretty much any processed foods and start to eat whole foods that you prepare yourself.
The reason it works so well and is so popular is because you see results almost immediately and that gives people the incentive to continue rather than waiting 6-8 weeks to see any difference.
But you need to know what you are doing before you do it. And frankly, I'm not as militant as many Atkins dieters in that I think you should use whatever book you want and follow it so that you will succeed.
If doing Induction for 3 weeks or 4 weeks works, do that. If you go a little over 20 carbs a day, don't worry. If you are less than 20 carbs a day, don't worry. Just read the book so that you know what you're doing and don't listen to the articles that tell you it's all about eating a pound of bacon and that it doesn't work. It DOES work for most people.