Uniqueness is not a "multi-axis affair". You have acknowledged what we have told you many times now, but you don't seem to have understood it. All that matters is how the password is generated. I could use any set of 4096 distinct tokens and have exactly the same password entropy as any other set, as long as each token is equally likely to be chosen. Each token could be a different word, or each could be a different spelling of the same word, or each could be a different integer, or each could be a string of n copies of the letter 'x'. It makes no difference in terms of security.
Of course, while all of the above are equally secure, that doesn't mean they are equally good choices of passwords for practical use. Passwords have other criteria of interest, such as length, memorability, and conformity to requirements. Using common words like Randall does is good for memorability but bad for length and conformity to typical password requirements (which often require non-letter characters, for instance). Using integers is bad for memorability, good for length, but still bad for conformity to requirements. Using different spellings of the same word is ludicrous and bad on all accounts. And using strings of copies of the letter 'x' is clearly the worst of them all. These are all equally secure, but they are not equally good.
Passwords used in a password manager are not required to be memorable, so other things are emphasized. Typically, they are randomly generated strings of characters drawn from a set which is small enough to be allowed by virtually all sites but large enough to have sufficient entropy for the given length. For these, there is no reason to do anything else; this system cannot be improved upon. Rejecting certain passwords because they double letters or whatever only reduces the space of potential passwords and therefore reduces entropy. But passwords we must memorize and use frequently should probably do something like Randall's recommendation. His system probably can be improved upon, but it is not obvious how, and it is already pretty good. One major downside is that many sites will not accept passwords consisting only of letters and spaces, but this can easily be solved by adding the same non-letter character to every password (0, for instance), which has no effect on entropy or memorability.
Your system is bad for memorizing, bad for satisfying requirements, and bad for security. You have already said the space of your passwords is (or technically has a bijection to) a strict subset of the space of Randall's passwords, meaning it has less entropy. And you have conceded that they are hard to remember. The advantage you claim is that they can be ordered according to an index, which I cannot really see the point of. In any case, the same is clearly true of Randall's. Whatever set of common words you are using, they will be listed in some particular order, so you can just number them starting at 1. Then, for any particular password, you can just replace all the words with their corresponding numbers (padded with zeroes) and concatenate them, giving a unique integer for each password and vice-versa. For instance, if the particular list you are using lists "correct" as the 107th word, "horse" as the 2451st, "battery" as the 74th, " and "staple" as the 3012th,
Needless to say, there are plenty of other ways you could choose to go about it, and they are all quite straightforward.