"Extermino" comes from "ex" (out of, out from) and "terminus" (border, boundary, demarcation). It means "to drive out" or "expel". The same document that talks about "exterminating" heretics also discusses receiving "exterminated" but repentant heretics back into the fold. To translate "extermino/-are" into English as "exterminate" without explanation is translational malfeasance.
Halsall knows this, or, as a historian and scholar, ought to know this. He just doesn't care and is happy to make the Church look bad. That a supposedly "Jesuit" school like Fordham tolerates him tells you what you need to know about them.