As suspected, there's absolutely nothing along the lines of what is claimed, that Christ gave "all believing Christians warrant to adjure the spirit of evil." [What does "the spirit of evil" mean here? Is that the same thing as an evil spirit? This author uses language very loosely. St. Thomas speaks of "demons" and not some vague "spirit of evil" as described by NewAdvent].
St. Thomas is merely asking where it is permitted to adjure demons period (and also men and also irrational creatures).
He first asks whether it is permitted to adjure other men. Here he makes the familiar distinction, using a slightly different word, "deprecatory" vs. "compulsory" ("of compulsion"). He states that it is permitted a superior to do use the compulsory (called by St. Alphonsus "imperative"), but otherwise it can only be deprecatory. Adjuration is an interesting question, with St. Thomas likening it to imposing an oath on someone else. Normally, an inferior is required to submit to a superior anyway, so I imagine that using the name of God somehow makes it more solemn? Does it render the obligation a mortal sin when adding the name of God into it? Neither St. Thomas or St. Alphonsus really delve into the matter, and this is just an aside.
So, I'll come back to the second question, the one under consideration here, about whether it is permitted to adjure demons, but what's interesting is in the third question, whether it's permitted/possible to adjure irrational creatures, St. Alphonsus disagree with St. Thomas. St. Thomas says yes, while St. Alphonsus says no. But that's not important here.
With regard to whether it is permitted to adjure demons, he begins with some objections, as per his usual approach, and then addresses them. 1) Origen says no, since it's a Jєωιѕн custom but not a power granted by Our Lord. 2) If it's permitted for us to adjure demons, then it would be permitted also for necromancers to apply various incantations to do the same. 3) In adjuring, you're establishing some association with the adjured, and it is not permitted to associate with demons. These seem like pretty weak objections, softballs for St. Thomas to knock out of the park.
So in response, he cites the Gospel of St. Mark where Our Lord says that "they shall cast out demons in My Name." And St. Thomas says that casting demons out is a form of adjuration (this will come into play later).
He says that it's permitted to adjure them out of compulsion, but not formulated as a request, since making a request would in fact be a problem according to objection 3, where you're establishing a relationship or association with them. But he says that compulsory adjuration is permitted, since demons are enemies. He'll also later say that even with compulsory adjuration, it's permitted to thwart their attacks, but not to get them to do other things, and not to learn things from them.
So compulsory adjuration is permitted against demons and only to repel their attacks, and not to obtains various benefits or knowledge from them (except an occasional rare case of a saint who's inspired to do something for the glory of God).
So, responding to the objections --
1) Origen was speaking about getting the demons to provide benefits to them, and not about repelling them.
2) Same thing goes for the necromancers with their incantations; that's forbidden because it's intended to obtain benefits/favors from the demons.
3) Repelling demons is not the same thing as forming a bond or relationship with them somehow.
AT NO POINT does St. Thomas discuss who can adjure demons, with what authority, and nowhere does he discuss the difference between solemn adjurations (by ministers using approved rites) and private ones. He is simply dealing with the question of whether it's permitted IN PRINCIPLE to cast out demons (and how, i.e. compulsion to repel attacks but not requests for assistance). There's absolutely nothing there along the lines of what NewAdvent had claimed, that he says that Christ gave "all believing Christians warrant to adjure the spirit of evil".
Here's an English translation (in addition to the Latin above) --https://www.ccel.org/ccel/aquinas/summa.SS_Q90_A2.html
He doesn't even explicitly state that someone must be a Christian to adjure demons in the name of God or of Christ, even though that should be obvious. He's simply talking about whether it's permitted in general or in principle.
What neither Doctor says is whether it is POSSIBLE to adjure demons to do something. They both agree that it's gravely illicit, but they don't discuss whether invoking the name of the Lord could compel them to do other things besides stop afflicting someone. They SEEM to imply that the power given to the Church was simply to get rid of them, but never explicitly say so. At one point, St. Thomas does say that the demons are subject to God's authority, and then implies that God communicated not all of His authority over demons, but just the ability to cast them out. That is also where the former dispute in St. Alphonsus comes into play, i.e. whether in asking the demons questions, etc., you're really acting within the confines of your actual authority or making more of a request (as the one author held who stated that it was a mortal sin to ask demons questions other than what's directly related to expelling them).
So it seems that the Church does not have authority over demons categorically but just vis-a-vis their expulsion.
AND, and this is crucial for us here, when you do NOT have authority to make a demand, the adjuration takes on the form of a request, which is grave sin and can be extremely dangerous. And that is why if there's any doubt whatsoever about whether we have the authority to adjure demons, it must be avoided, because it can backfire and result in retaliation.