All you have to do is hit dictionary.com to find out how wrong you are.
Ex*ter"mi*nate\, v. t. [imp. & p. p. Exterminated; p. pr. & vb. n. Exterminating.] [L. exterminatus, p. p. of exterminare to abolish, destroy, drive out or away; ex out + terminus boundary, limit. See Term.] 1. To drive out or away; to expel.
The dictionary disagrees with the almighty and allknowing Havoc. Nice try, though.
The same document from Lateran IV goes on to describe what to do about "exterminated" heretics who repent.
Exterminate: primary meaning listed: "To get rid of by destroying completely; extirpate. See Synonyms at abolish."
The princton notation has a singular purpose of mind pretty much - kill off or utterly destroy. Rome wished to eliminate them - which is why they pushed for the death penalty, invoked it at Rome pretty much simultaneous to the constitution of Lombardy, then began burning heretics at the stake. It isn't merely a matter of what your document says - it's a matter of what it says in context with what your clergy's actions were and their other writings. The document stands easily on it's own. Rome didn't set about merely running heretics off Catholic lands. They rather seized the lands of heretics, tried, convicted and burnt them at the stake. That is, of course, after in many cases the application of Ad Exterpanda where they first solicited confessions from people by doing the likes of pouring molten led down their throats. Whether they could confess after this sort of torture or not, it really didn't matter if they were guilty - did it. Run them off huh.. in a pig's eye. Plain and simple. Rome has claimed it wasn't their fault - it is nothing if not their fault.