I don't know why "boolits" got started unless it was making fun of imagined hick accents from some fictional part of the country? Maybe Li'l Abner said it that way? We have a (just barely) phonetic writing system of written English, and I don't know anyone who actually pronounces the word the way that spelling suggests? Of course, I know few people who pronounce the current Russian president's name correctly, either.
Anyway, cast lead bullets had the name before jacketed bullets were made, so I think it has dibs on the original terminology. It is jacketed bullets that should receive the new name, if there is to be one? Stuffed Jackets? Don't know what we'll do with solids, in that case?