Thomas Aquinas had no great difficulty with the definition.
Aquin.: SMT SS Q[11] A[3] Body Para. 1/2
I answer that, With regard to heretics two points must be observed: one, on their own side; the other, on the side of the Church. On their own side there is the sin, whereby they deserve not only to be separated from the Church by excommunication, but also to be severed from the world by death. For it is a much graver matter to corrupt the faith which quickens the soul, than to forge money, which supports temporal life. Wherefore if forgers of money and other evil-doers are forthwith condemned to death by the secular authority, much more reason is there for heretics, as soon as they are convicted of heresy, to be not only excommunicated but even put to death.
Aquin.: SMT SS Q[11] A[3] Body Para. 2/2
On the part of the Church, however, there is mercy which looks to the conversion of the wanderer, wherefore she condemns not at once, but "after the first and second admonition," as the Apostle directs: after that, if he is yet stubborn, the Church no longer hoping for his conversion, looks to the salvation of others, by excommunicating him and separating him from the Church, and furthermore delivers him to the secular tribunal to be exterminated thereby from the world by death. >For Jerome commenting on Gal. 5:9, "A little leaven," says: "Cut off the decayed flesh, expel the mangy sheep from the fold, lest the whole house, the whole paste, the whole body, the whole flock, burn, perish, rot, die. Arius was but one spark in Alexandria, but as that spark was not at once put out, the whole earth was laid waste by its flame."
Of course Aquinas wasn't in spin mode.
Sounds to me like a good justification for the death penalty. Follow due process, but if the SOB is hardened to his crime, then kill him. Which goes back to a point of the article. The Inquisition introduced due process, and a high level of dueprocess into the prosecution of heretics. Of course, you can't admit that heresy could be a crime. This is a bit like liberals who could never admit that being a communist could be a crime. But I won't push the matter. In any case, the argument is made in the article that a person accused of heresy was more likley to receive what we would call a fair ttrial than a person accused of, say, theft. I think that is the chief claim of the article, that the Inquisition provided for a fair trial, at least in theory and often also in fact. A lawyer's trick: man accused of stealing a pig claims that his religious beliefs allowed him to do it. A matter of heresy. The case is taken over by the Church court. He adjures his heresy and gets off with a penance. This is the practical side of an ecclessiastical court being more lenient than a secular court. For this reason everyone one who could read and write claimed clerical status.
If Thomas Aquinas had understood "to exterminate" to mean "to kill," why would he bother to add "from the world by death?"