I think you're right about the differentiation between groups. These elemental attributions definitely come from the Ciceros, to Donald Kraig, to Damien Echols.

As for Raphael having a sword, I believe it is connected to Tarot suits. The suit of Swords is connected to the air element, so Raphael has a sword. The suit of Wands is associated with fire, so Michael has a wand/staff.