"Blood mages" in WC3 practiced magic that had nothing to do with blood. They casted Flamestrike, Banish, Siphon Mana, and Summon Phoenix. They were magi with a darker feel than the those of Dalaran - consistent with the blood elves' theme of turning to darker paths and darker magic due to their withdrawal from the Sunwell's magic (hence the warlock-y spells like Banish and Siphon Mana, as well as the more volatile fire mage spells like Flamestrike and Summon Phoenix). The "blood" in "blood mage" was more in the sense of "blood" in "blood elf": it did not have anything to do with manipulating blood; it was just a title-name-thing like "Sin'dorei".
As to how "blood mage" changed to signify a mage who manipulated blood magics, my guess is that either a new dev tried to bring in lore about them in game but was not knowledgable about what they were in WC3, or they tried to develop it beyond that.
Though if they wanted to develop blood mages as mages who manipulated blood, they probably should have appropriated it to the San'layn, since they have a heritage of wielding magic and it plays well with their vampiric theme.
This should be programmed to be an automatic response to all blood mage/ blood magic threads.