I recommend calling up admissions offices and asking. Don't identify yourself if you don't want, just say "Hey, I'm applying this next cycle. I know you don't ask for one, but if I were to send in a diversity statement in addition to my personal statement, how would that reflect on my file/application?"
I did this and got several staunch, "don't even bother"s and even an insinuation that it would be a strike against my application.
And as a blanket statment, most people on this board are in the same position as you are in terms of reading the minds of LSAC and the adcomms...meaning we can't. If you have a question like that -- basically "what should I send the adcomms" -- call and ask them. They are used to taking questions. They know what they want come application time, and if they don't want 4 extra pieces of paper to sift through, they will tell you. It's as simple as that. While the criteria for getting in and the LSAT scale might be mysteries, the completion of applications is not a dark, clouded process on which we have to speculate.