You should think it like this:
1) Rehalogenating bleaches -- ALL of the silver is turned to silver halides. This is needed when the whole image is to be redeveloped. Example: Potassium ferricyanide + bromide.
2) Typical color bleaches, often called non-rehal -- not all silver is turned to silved halides; some is removed directly. But these are still PARTIALLY rehalogenating! Example: Ammonium ferric(III) EDTA
3) BW reversal bleach, which is completely non-rehalogenating. It attacks the emulsion quite a bit, is very acidic and more toxic and expensive and needs a clearing bath afterwards because it stains the emulsion, so there is no point in using it if the complete direct removal of silver is not needed. In BW reversal, this is a must because the existing halides are used to make the final positive image; if the bleach converted any silver to halides, they would mix together. Example: Sulfuric acid + permanganate or dichromate.
In E6, all of the silver is removed anyway so there is not a mix-up problem -- dyes are only created in the second development. First one is non-dye creating developer (BW developer)