Here is where the confusion lies.
You can do it EITHER way. You can do it 4 days in between or on the 4th day. There is no magic to the pattern, it is just an easy way of being in with one attempt effortlessly. The option you choose depends on what you personally want to accomplish - if you want to boost odds of conception, you have the attempt sooner, if you want better odds of pink, have it later. BOTH are still e4d method. It really isn't confusing, it isn't so much about the precise number of days, it's just having sex every 4ish days to be in with a chance of conception without having to mess around with OPK or guessing about when you're going to O.
There are a couple of reasons why people do this method. Understanding them will further shed light on what option you want to do.
One of them is for those who have been swaying a long time. They want to improve the chances they get pregnant and are ready to possibly compromise a bit in the swaying department (although this is still getting good results for pink provided people have been on diet 12 weeks or more) So they have sex every 4th day, so the worst case scenario is ending up with attempts on O-3 and O Day (you don't HAVE to or want to have attempts exactly on that day necessarily, just that this is the worst you can do, more likely it will be on O-2 or O-1)
The other is for people who are just starting off, and can't or don't want to use OPK. These people probably prefer a better chance of pink and can stretch it an additional day (or even more if they so choose) and are ok with slightly lower odds of conception to start out with. So they err on the side of fewer attempts and stretch it that one extra day in between.
There is no right or wrong way to do it. The point is NOT the magic pattern, because there is no magic pattern. The only thing that really seems to help is the one attempt. So we're trying to hit that one attempt without completely negating the odds of getting pregnant and no mattter how you do it, e4d (be it on the 4th day or with 4 full days between) is just a tool we use to accomplish the one attempt.

