Continued…
The Cathari are schismatics; but it seemed good to the ancient authorities, I mean Cyprian and our own Firmilianus, to reject all these, Cathari, Encratites, and Hydroparastatæ;, by one common condemnation, because the origin of separation arose through schism, and those who had apostatized from the Church had no longer on them the grace of the Holy Spirit, for it ceased to be imparted when the continuity was broken. The first separatists had received their ordination from the Fathers, and possessed the spiritual gift by the laying on of their hands. But they who were broken off had become laymen, and, because they are no longer able to confer on others that grace of the Holy Spirit from which they themselves are fallen away, they had no authority either to baptize or to ordain. And therefore those who were from time to time baptized by them, were ordered, as though baptized by laymen, to come to the church to be purified by the Church’s true baptism. Nevertheless, since it has seemed to some of those of Asia that, for the sake of management of the majority, their baptism should be accepted, let it be accepted. We must, however, perceive the iniquitous action of the Encratites; who, in order to shut themselves out from being received back by the Church have endeavoured for the future to anticipate readmission by a peculiar baptism of their own, violating, in this manner even their own special practice. My opinion, therefore, is that nothing being distinctly laid down concerning them, it is our duty to reject their baptism, and that in the case of any one who has received baptism from them, we should, on his coming to the church, baptize him. If, however, there is any likelihood of this being detrimental to general discipline, we must fall back upon custom, and follow the fathers who have ordered what course we are to pursue. For I am under some apprehension lest, in our wish to discourage them from baptizing, we may, through the severity of our decision, be a hindrance to those who are being saved. If they accept our baptism, do not allow this to distress us. We are by no means bound to return them the same favour, but only strictly to obey canons. On every ground let it be enjoined that those who come to us from their baptism be anointed in the presence of the faithful, and only on these terms approach the mysteries. I am aware that I have received into episcopal rank Izois and Saturninus from the Encratite following. I am precluded therefore from separating from the Church those who have been united to their company, inasmuch as, through my acceptance of the bishops, I have promulgated a kind of canon of communion with them.
newadvent.org/fathers/3202188.htm
Now, as far as acceptance of baptism is concerned, Basil distinguishes between three groups: heretics, schismatics and unlawful congregations. While he rejects the baptism of heretics, he admits the baptism of schismatics, saying that they still “belong to the Church.” Moreover, “heresy” seems to be used in a limited sense such that, the Catholic understanding of the papacy, to give an example, would not constitute heresy, which Basil defines as “disagreement concerning the actual faith in God.” Nor does “acceptance” appear to point to merely “acceptance by oikonomia.” Basil rejects heretical baptisms on the basis of their fundamentally errant faith in God. To put this in perspective, he is speaking of various Gnostic groups and the Montanists. If Basil thought schismatic baptisms were invalid but should be accepted through economy, he could have simply said heretical baptisms were rejected because they were not in communion with the Catholic (Orthodox) Church. Rather, he argues that Montanist baptisms are invalid because when they use the Trinitarian formula, they intend to baptize in the name of the Father, and the Son and Montanus or Priscilla, i.e. they disagree about actual faith in God.
Interestingly, it seems that Basil would accept current-day Catholic baptisms as valid, unless we take a very hard-line stance of the Filioque issue, saying that it is a fundamentally distorted view of God. In any case, it does not appear that Basil agrees with the Orthodox position expressed in the thread, nor does he agree with Cyprian and Firmilianus.
I found this article, which appears to endorse my reading, more or less.
myriobiblos.gr/texts/documents/northamerican_4.htm
Not being really up to speed on history, I don’t intend to argue this point any further than I have, but I think you might want to reconsider whether Basil’s position was not more nuanced than you have implied.
When Augustine is speaking about sacraments outside of the Church, I would assume he is speaking of (valid) sacraments outside the visible boundaries of the Church. I’m sure he would admit that wherever baptism is present, the Church is also present. Maybe his work On Baptism deals with this. Have you read it? I don’t think it is remotely tenable to limit the grace of God to the visible boundaries of the Church.