I think weapon type is based on spawn location. Might also be mission type.
I've gone through four Lichs so far;
Telesto (Exterminate) spawned a Lich with Quartakk
Cassini (Capture) spawned a Lich with Karak, twice.
Numa (Rescue) spawned a Lich with Drakgoon
Can someone prove this wrong?
Edited to include new lich. (orignally 3)