Fieldcraft Demimateria comes from DoL/DoH tools/armor/accessories.

Battlecraft Demimateria comes from DoW/DoM weapons/shield/armor/accessories.

They become more frequent when breaking item level 30+ things, but I got my 1st one from a lv26 tool.

It may have been just my luck, but green icon gear from dungeons usually yielded Battlecraft Demimateria for me. 2 pieces from 1 green icon item and it was item level 30-ish.