Red because Arkana has no animated cutscene if he summons Dark Magician unless it is the Red version so it stands to reason it has to be that specific version. Tho I feel they should have had Hugo's playmate be purple and not black. Yugi did not have a Black Dark Magician.
You do not need to play using Arkana when summoning the Red DM, any character will do... I summon Red DM using both Yami Yugi and Arkana during the Battle City event and I got my Red DM mat...