I have all the Atkins Diet Books, plus other books he wrote like "age-defying" and "vita-nutrient" etc. purchased at thrift stores and library book sales over the years, a sort of Atkins through the ages.
As Jo mentioned, there have been many tweaks along the way, but consider "just Atkins" as using Real Food..no Bars, shakes, or frozen meals as pushed on the website.
One caveat about even the first book you have, is if he had a patient in his clinic with diabetes or who needed to lose significant weight, they would be encouraged to stay on Induction as long as possible. When Dr Westman first visited the Atkins clinic to design a clinical trial and the diet rules to be used for it, many patient records showed that. So the trial Diet (still used at Duke today) is basically the first Induction week until at goal, no added foods, no phases, etc. Books are written to address the diet needs of everyone even if they want to lose 10 pounds; clinic plans help those with serious health and weight issues. The diet is further personalized in medical check-ups as weight is lost, but for a good start, using the edition you bought and sticking to the first week rules works well.
Edit: After I wrote this, checked our library's "on order" list, apparently the Atkins company has a new book in the works,
Atkins: Eat Right, Not Less: Your Guidebook for Living a Low-Carb and Low-Sugar Lifestyle Appealing to those who won’t go the NGNS (no grain, no sugar) road.