The consensus of the Reformation is that the censure of (greater) excommunication is reserved for the obstinate or stubborn sinners, and should not be used on those that profess repentance or sorrow when admonished by the session or presbytery. However, they can be suspended from the Lord's Supper for a time (sometimes called "lesser excommunication"), in order to have an opportunity to discern the sincerity of their repentance.
That is what the Book of Discipline is saying.
Note that I am expressing a matter of consensus among the Reformed Orthodox here, and not just my personal opinion.
Junius, Ames, and Walaeus explain the distinction well:
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Junius: "Excommunication is two-fold: lesser and greater. Lesser excommunication is that by which those who wickedly run up against the faith, or good morals, or public order, are excommunicated; not so that they cease in the absolute to be reckoned as members of Christ; but rather, so they are barred from the Lord’s Supper only for a time, until their penitence, or their faith and dutifulness, is judged sufficient, and a remedy is provided for public scandal. And these men were called the”abstenti” by the ancient writers. See Cyprian, in a letter of his.
Greater excommunication is the sacred action of a presbytery, by which, according to the rule taught by Christ, a brother who is obstinate in his sins is separated from the communion of believers, and, being cast out of the Church, he is handed over to Satan, so that, being punished, he repents, and having demonstrated his repentance, he is received again. And under the Old Testament, this was “to be put out of the synagogue” (ἀποσυνάγωγον γένεσθαι), Jn. 9:22, 12:42, 16:2. And thus, in the ancient Church, excommunication was in four distinct degrees, the first of which was called standing (σύστασις); the second, falling down (ὑπόπτωσις); the third, hearing (ἀκρόασις); and the fourth, crying in front [of the church building] (πρόκλαυσις). For, the canons distinguish them in this way, in which they are called those who stand (συνιστάμενοι), those who fall down or submit (ὑποπίπτοντες), those who kneel (γονυκλίνοντες) (Can. 15 of the Council of Caesarea), hearers (ἀκροώμενοι), and those who cry out front (προκλαίοντες). The first ones pertain to lesser excommunication, and the rest to greater excommunication."
Ames: "Full excommunication is not to be employed unless obstinacy is added to the sin. Mt. 18:17. For the sinner that has been rightly admonished must necessarily show himself either repentant or obstinate. Moreover, the penitent one is not to be excommunicated. Therefore, only the obstinate one.
Nevertheless, in more atrocious, scandalous sins, such great patience and delay is neither necessary nor useful, in order to wait for repentance or discern obstinacy as in more typical sins. When the matter itself allows for delay, it is agreeable to Scripture and reason for excommunication to first be begun by suspension or abstention from the supper, and similar privileges of the Church, which tends to be called lesser excommunication. Nevertheless, one should not dwell on this degree; but repentance is to be insisted on in this manner and in this space of time; and when there is no longer any hope of it, one must finally proceed to a complete separation from the fellowship of believers, which tends to be called greater excommunication.
Moreover, because an obstinate sinner cannot be separated from believers, unless believers are also separated from him, and because this also serves for their saving shame, 2 Thes. 3:14, on this account, those who are lawfully excommunicated are to be shunned by all communicant members, not as to duties that are simply moral, or otherwise necessary, but as to those parts of interaction which tend to go along with approval and a more intimate familiarity. “Conversation, prayer, greetings, feasts, and shared meals are denied.”"
Walaeus: "Concerning the nature of lesser excommunication, it is asked, Does it have a foundation in Scripture? Certainly, it is without controversy that it was in common use in the early church, because the “abstenti” and the penitent were distinguished from those who were excommunicated, and they had a particular place in the Church, as is known. Also, the reason is obvious: because, even if someone, after a grave sin and scandal committed against the Church, professes repentance, nevertheless, it must be proven to the Church whether that repentance is sincere, which must be gathered from its fruits. So that, in the meantime, the scandal is removed from the offended members of the Church, and just as he did with his sin, so also, in the meantime, he makes his repentance known to others. Nevertheless, we acknowledge that, in establishing the time for penitence, the ancients were more strict than the apostolic custom appears to require. But, beyond this, some foundations must also be given for this discipline of the Church.
I. The first is sought from the abstenti in the Old Testament, who, on account of ceremonial or moral impurity, were compelled to abstain from the common sacrifices, and sacraments, until they were cleansed, and reconciled by particular signs and sacrifices of repentance. Yet, they were not therefore cast out of the Church, or uprooted from the people.
II. The passages 1 Cor. 5 and 2 Cor. 2 and 7, compared to each other, evince it. For Paul judges that that incestuous man is to be excommunicated, but he repents in anticipation of this, and they ask the Apostle what is therefore to be done by them. And to this question, he responds, in the second epistle, that the sinner must not be afflicted with such great sorrow that he is consumed; but the ἐπιτίμησις, the rebuke, done by many is sufficient; and that, therefore, he wishes for him to be received again by the Church. Now, reasonably, in the meantime, he professed his repentance; and nevertheless, he was doubtlessly kept [from the sacrament], until they had received a response from the Apostle.
III. From the passage 1 Cor. 11:28, “But let a man examine himself, and so let him eat of that bread, and drink of that cup.” If, therefore, it is the responsibility of individuals to examine themselves, so that they do not eat unworthily, therefore, it is also the responsibility of the rulers of the Churches, to prove others, so that they do not eat unworthily, and eat or drink judgment upon themselves, or with scandal to others, as is said in the same place, at the end of the chapter [1 Cor. 10:32]: “Give offense to none, neither to the Jews, nor to the gentiles.”
IV. Some also think that lesser excommunication is indicated in the words of Christ, Mt. 18:17, when he says, “Let him be to you as a publican.” A publican could pray and be in the temple, Lk. 18:13, even though they could not share in sacrifices or sacraments. Nevertheless, Beza, against Erastus, judges the contrary, and that this phrase comes from the fact that a publican was more reviled than a Gentile. Moreover, they could not be shut out of the temple because they were public servants of the Romans, and Jews, who they did not dare to shut out. Or they were in the outermost part of the temple, where even Gentiles could appear, from 1 Kng. 8:41. Nevertheless, afterward, the Court of the Gentiles appears to have been distinguished under Herod from the Court of Israel.
V. 1 Cor. 10:21, “You cannot be partakers of the Lord’s table, and of the table of devils.” Also 1 Cor. 5:11, “Do not eat with such a person.” Therefore, much less is it lawful to grant them the Supper. 1 Cor. 10, the Apostle addresses that which cannot be done rightly and with a good conscience, and therefore is wicked.
What is objected, then, is that God immediately forgives the one who repents; and therefore, the Church must do the same.
Resp. This is true, when it is clear to the church; and also, when it can be done with edification; for the Church is not a searcher of hearts, and it must also have regard for the scandal or edification of the weak. Rom. 14:13.
1. It is objected by Erastus that all who were circumcised were able to celebrate the Passover with others; and thus, it is enjoined upon every head of household, Ex. 12:4. Therefore, also the Supper.
Resp. The antecedent is denied, because those who were impure were, according to the law, compelled to abstain for a whole month, in order to cleanse themselves in the meantime. Num. 9:10; 2 Chron. 30:17.
2. Obj. If it is not lawful for any Christian to separate himself from the Supper without the sin of schism, on account of the vices which rule over many in the Church, as long as the doctrine is pure; therefore, neither is it lawful for the Church to separate anyone from the Supper, as long as his confession is pure.
Resp. The consequence is denied: for the nature of a private man is one thing, and that of the whole Church, or of the rulers of the Church, another. For they are commanded to watch that the sacraments are not profaned, and to make a distinction between what is pure and impure, with public authority, which it is not lawful for individuals to take upon themselves: for the former is a remedy against sins, but the latter violates charity.
3. Christ, 1 Cor. 11:28, commands for all to eat of that bread, and drink of this cup, as long as they have proven themselves.
Resp. But if they do not prove themselves, and yet they wish to eat of this bread, and drink of this wine, nevertheless, it is not to be permitted them, if the matter is obvious, so that they do not eat harsher judgment upon themselves, as we heard concerning the impure in the Old Testament.
4. 1 Cor. 10:2-4 says that the fathers “all ate the same spiritual food, and they all drank the same spiritual drink,” which corresponds to the Supper. But many of them were ungodly. Therefore…
Resp. The nature of these things is different, because here, the food and drink was not only sacramental, but also corporeal. Secondly, God punished them. Thirdly, their great number made them more negligent in this.
5. But there were many drunks, fornicators, and idolaters in the Corinthian Church, and yet he does not command them to abstain from the Supper.
Resp. Even if he does not command it everywhere, nevertheless, he commanded it in 1 Cor. 5:4. For the one who commands what is greater also commands what is lesser. Secondly, he who forbids them from eating food with them much more forbids them from partaking of the Supper. Finally, he threatens in 2 Cor. 10:6 that he will come to them with a rod unless they do their duty.
It is called greater excommunication, when one who is called a brother, on account of grave impurity of doctrine or life, having shown contempt for the private and public admonitions of the Church, is still obstinate in his sin, and he is pronounced to be shut out, not only from the Holy Supper, but also before the whole Church, in God’s name, from the fellowship of the Church, by a public sentence on earth, and consequently, in heaven, until such a time as he is reconciled to God and the Church by true and sincere repentance. This excommunication is also called “handing over to Satan” by the Apostle, 1 Cor. 5:5; 1 Tim. 1:20, because he who is cast out of the Church is cast into the kingdom of Satan, who reigns outside Christ’s Church. Even though Chrysostom and Augustine thought that, in the first Church, a man of this sort was also handed over to Satan to be tormented bodily. The Apostle, in the same place, calls it “the cleansing of leaven,” and “removal from the Church’s midst.” Christ, on the other hand, Mt. 18:17, says “let him be to you as a heathen and a publican;” namely, a man with whom one has no fellowship in sacred things; that is, respecting all the signs of grace which God gives his Church.
In this definition, some of its parts are to be explained, besides those that were explained above, concerning its instrumental cause and subject; and that against the Mennonites, who excommunicate for any reason, and who split their Churches, so that they appear to follow the example of a house of buyers and sellers. And, I. it is certainly employed, not on account of the routine infirmities of men, but only with those who gravely sin against the doctrine or commandments of God, as is explained in 1 Cor. 5:5; sins on account of which men are shut out of the kingdom of God, and which are a great scandal to the Church, as in Gal. 5:12, “I would that they were even cut off which trouble you.” Also Rom. 16:17; 1 Tim. 1:20.
It is asked, Should it be used on account of light matters? In this way, the bishops excommunicate on account of a debt of a small sum of money, and contracted without fraud, when, nevertheless, men of this sort can be godly. 2 Kg. 1:8.
Therefore, we respond that routine sins do not appear to be punished this way in the Scriptures, unless they become grave sins, and liable to scandal, by the addition of obstinacy and contempt. They are purged by routine, ordinary repentance. But rather, it is used for those for which extraordinary, open repentance is necessary for their expiation. Nor is it even used for serious, hidden sins, as long as they remain hidden, as is clear from Mt. 18:15. Therefore, it is only used with sins that are serious, and open, and for which there is need of open, extraordinary means, in order to lead a man to repentance.
II. Not even in serious or the most serious sins, unless lasting obstinacy and rebellion is added to it, contrary to all admonitions, private and public.
And this is stated in opposition to the ancient Novatians, and the recent Mennonites, who in sins of this sort first excommunicate the man, and then admonish him. For they wish for him to repent outside of the church, not within the Church, because of 1 Cor. 5:1, where Paul does not precede with any admonition; but he simply wishes for the incestuous man to be excommunicated. But we respond, this passage is not to be considered alone; instead, it must be compared with others. For, in 2 Cor. 2:6 and 7:9, the Apostle wishes for him to be forgiven after an admonition, or a “rebuke by many.” Therefore, an admonition preceded, while he was not yet excommunicated. 2. Christ’s words in Mt. 18:17 are clear: “If he neglect to hear the church.” Also Tit. 3:10."
