Actually, no.
The evidence (and it mostly comes from St. Stephen’s) opponents was that it involved all of Africa and the East versus Rome. The canon of the Ecumenical Council did not adopt either extreme, accepting some baptisms of some heretics, but not others. Neither Stephan nor his opponents made a distinction between heretics on the matter.
See Eusebius, and the notes:
ccel.org/ccel/schaff/npnf201.iii.xii.iii.html
ccel.org/ccel/schaff/npnf201.iii.xii.v.html
ccel.org/ccel/schaff/npnf201.iii.xii.vi.html
There is another section in Eusebius in which the Popes of Alexandria (btw, this is the earliest attested use of the term “Pope” for a specific hierarch) that the practice of Cyprian was of long standing there:
ccel.org/ccel/schaff/npnf201.iii.xii.viii.html#iii.xii.viii-Page_296
I don’t know why you cited c. XIX of Nice I as “sure did,” as the canon only speaks of rebaptism:
Concerning the Paulianists who have flown for refuge to the Catholic Church,
it has been decreed that they must by all means be rebaptized; and if any of them who in past time have been numbered among their clergy should be found blameless and without reproach,
let them be rebaptized and ordained by the Bishop of the Catholic Church; but if the examination should discover them to be unfit, they ought to be deposed. Likewise in the case of their deaconesses, and generally in the case of those who have been enrolled among their clergy, let the same form be observed. And we mean by deaconesses such as have assumed the habit, but who, since they have no imposition of hands, are to be numbered only among the laity.
ccel.org/ccel/schaff/npnf214.vii.vi.xxviii.html
The same cite gives the quote of Arles:
If anyone shall come from heresy to the Church, they shall ask him to say the creed; and if they shall perceive that he was baptized into the Father, and the Son, and the Holy Ghost, he shall have a hand laid on him only that he may receive the Holy Ghost. But if in answer to their questioning he shall not answer this Trinity, let him be baptized.
St. Stephen doesn’t seem to have made this distinction.
Of course, arent we discussing Cyprian’s disagreement with Rome? That is why I have cited these specific canons. Hefele expounds the disagreement well. Allow me to quote from a post that I delivered in another forum:
Let us first turn to the Council of Arles(314). Before I quote the canon associated with re-baptism let us first turn to St. Augustine:
“The question relating to re-baptism was decided against Cyprian,
in a full council of the whole Church” (Hefele, His. Coun.).
The more important question remains, what council was the saint refering to? Either the Council of Arles or 1st of Nicea are the options. Either way, he views it as opposing Cyprian’s stance.
Hefele notes on the 8th canon of Arles(314):
“We have already seen that several African synods, held
under Agrippinus and Cyprian, ordered that whoever had been
baptized by a heretic, was to be re-baptized on re-entering the
Church. The Council of Arles abolished this law of the
Africans, and decreed that those who had received baptism
from heretics in the name of the holy Trinity were not to be
again baptized, but simply to receive the imposition of hands,
ut accipiat Spiritum sanctum. Thus, as we have already said,
the imposition of hands on those converted was ad paenitentiam
and ad confirmationem. The Council of Arles promulgated in
this eighth canon the rule that has always been in force, and
is still preserved in our time, with regard to baptism conferred
by heretics: it was adopted and renewed by the nineteenth
canon of the Ecumenical Council of Nicea.”
Thus St. Vincent of Lerins states in the 6th chapter of his Commonitory:
"Great then is the example of these same blessed men, an example plainly divine, and worthy to be called to mind, and medirated upon continually by every true Catholic, who, like the seven-branched candlestick, shining with the sevenfold light of the Holy Spirit, showed to posterity how thenceforward the audaciousness of profane novelty, in all the several rantings of error, might be crushed by the authority of hallowed antiquity.
Nor is there anything new in this? For it has always been the case in the Church, that the more a man is under the influence of religion, so much the more prompt is he to oppose innovations. Examples there are without number: but to be brief, we will take one, and that, in preference to others, from the Apostolic See, so that it may be clearer than day to every one with how great energy, with how great zeal, with how great earnestness, the blessed successors of the blessed apostles have constantly defended the integrity of the religion which they have once received.
Once on a time then, Agrippinus, bishop of Carthage, of venerable memory, held the doctrine—and he was the first who held it—that Baptism ought to be repeated, contrary to the divine canon, contrary to the rule of the universal Church, contrary to the customs and institutions of our ancestors. This innovation drew after it such an amount of evil, that it not only gave an example of sacrilege to heretics of all sorts, but proved an occasion of error to certain Catholics even.
When then all men protested against the novelty, and the priesthood everywhere, each as his zeal prompted him, opposed it, Pope Stephen of blessed memory, Prelate of the Apostolic See, in conjunction indeed with his colleagues but yet himself the foremost, withstood it, thinking it right, I doubt not, that as he exceeded all others in the authority of his place, so he should also in the devotion of his faith. In fine, in an epistle sent at the time to Africa, he laid down this rule: Let there be no innovation—nothing but what has been handed down. For that holy and prudent man well knew that true piety admits no other rule than that whatsoever things have been faithfully received from our fathers the same are to be faithfully consigned to our children; and that it is our duty, not to lead religion whither we would, but rather to follow religion whither it leads; and that it is the part of Christian modesty and gravity not to hand down our own beliefs or observances to those who come after us, but to preserve and keep what we have received from those who went before us. What then was the issue of the whole matter? What but the usual and customary one? Antiquity was retained, novelty was rejected.
But it may be, the cause of innovation at that time lacked patronage. On the contrary, it had in its favor such powerful talent, such copious eloquence, such a number of partisans, so much resemblance to truth, such weighty support in Scripture (only interpreted in a novel and perverse sense), that it seems to me that that whole conspiracy could not possibly have been defeated, unless the sole cause of this extraordinary stir, the very novelty of what was so undertaken, so defended, so belauded, had proved wanting to it. In the end, what result, under God, had that same African Council or decree? None whatever. The whole affair, as though a dream, a fable, a thing of no possible account, was annulled, cancelled, and trodden underfoot.
And O marvellous revolution! The authors of this same doctrine are judged Catholics, the followers heretics; the teachers are absolved, the disciples condemned; the writers of the books will be children of the Kingdom, the defenders of them will have their portion in Hell. For who is so demented as to doubt that that blessed light among all holy bishops and martyrs, Cyprian, together with the rest of his colleagues, will reign with Christ; or, who on the other hand so sacrilegious as to deny that the Donatists and those other pests, who boast the authority of that council for their iteration of baptism, will be consigned to eternal fire with the devil?"
As far as the more ancient custom goes, Cyprian himself tries to relate that the re-baptism of heretics is of no innovation. However, Cyprian , just as St. Vincent, attributes the earliest he knows of the custom to Agrippinus; which is to say in the early 3rd Century (possibly 220 A.D.) He only replies:
“Is antiquity, then, more precious than truth?”
Moreover, he states:
“In spiritual things we must observe what the Holy Spirit has (afterwards) more
fully revealed (id in spiritualibus sequendum, quod in meliiis
fuerit a Spiritu sancto revelatum).” He acknowledges, therefore,
in his practice a progress brought about by the successive
revelations of the Holy Spirit (Hefele, Hist. Coun.).
If it hasnt become clear yet that Cyprian acknowledges the older custom of not re-baptizing; he further states:
“Divine mercy may well come to their aid; but because one has
erred once, it is no reason for continuing to err (non tamen, quia
aliquando erratum est, idea semper errandum est).”
But what was Firmilian’s stance on the antiquity of the two customs? He writes in a letter documented with the Letters of Cyprian:
" You Africans, can answer Stephen, that having found
the truth, you have renounced the error of your (previous) custom
(vos dicere Afri potestis, cognita veritate crrorem vos consududinis reliquisse)."
Firmilian acknowledges the older custom, but wishes to attribute it to purely human tradition. But how did Firmilian respond when he felt the need to defend his territory(Asia Minor) against his adversaries? He responds with the greatest answer of all: “We do not remember when this practice began amongst us!” He finally appeals to the synod of Iconium, which is to say, about the year 230 A. D.