The weapon's appearance never changes the damage type. And there is actually a third one, shot, which is for Bowguns and Bows.
The only impact attack the SnS has is the shield bash, all other hits do cutting. Switch Axes with Exhaust Phial can KO with attacks to the head like Hammers and the aforementioned shield bash, but they still do cutting. Same deal with Punish Draw.
TL;DR: damage type depends on the attack itself, not which weapon of the class you use. --- http://www.backloggery.com/neslug Nintendo Network ID: Neslug