Well, I fully support the concept of Apostolic Succession, that's true, but it is a stretch to say that it is a long-held Catholic tradition that only baptisms performed as you stated are valid. In fact, the contrary is true:
From the Code of Canon Law:
Can. 861 §1. The ordinary minister of baptism is a bishop, a presbyter, or a deacon, without prejudice to the prescript of ⇒ can. 530, n. 1.§2. When an ordinary minister is absent or impeded, a catechist or another person designated for this function by the local ordinary, or in a case of necessity any person with the right intention, confers baptism licitly. Pastors of souls, especially the pastor of a parish, are to be concerned that the Christian faithful are taught the correct way to baptize.
Translations from Vatican-speak:
According to the Catechism of the Catholic Church:
1256 The ordinary ministers of Baptism are the bishop and priest and, in the Latin Church, also the deacon.57 In case of necessity, any person, even someone not baptized, can baptize, if he has the required intention. the intention required is to will to do what the Church does when she baptizes, and to apply the Trinitarian baptismal formula. the Church finds the reason for this possibility in the universal saving will of God and the necessity of Baptism for salvation.58
From the Catholic Encyclopedia (1907):
In case of necessity, baptism can be administered lawfully and validly by any person whatsoever who observes the essential conditions, whether this person be a Catholic layman or any other man or woman, heretic or schismatic, infidel or Jew.The essential conditions are that the person pour water upon the one to be baptized, at the same time pronouncing the words: "I baptize thee in the name of the Father and of the Son and of the Holy Ghost." Moreover, he must thereby intend really to baptize the person, or technically, he must intend to perform what the Church performs when administering this sacrament.
The Roman Ritual adds that, even in conferring baptism in cases of necessity, there is an order of preference to be followed as to the minister. This order is: if a priest be present, he is to be preferred to a deacon, a deacon to a subdeacon, a cleric to a layman, and a man to a woman, unless modesty should require (as in cases of childbirth) that no other than the female be the minister, or again, unless the female should understand better the method of baptizing. The Ritual also says that the father or mother should not baptize their own child, except in danger of death when no one else is at hand who could administer the sacrament. Pastors are also directed by the Ritual to teach the faithful, and especially midwives, the proper method of baptizing. When such private baptism is administered, the other ceremonies of the rite are supplied later by a priest, if the recipient of the sacrament survives.
This right of any person whatsoever to baptize in case of necessity is in accord with the constant tradition and practice of the Church. Tertullian (De Bapt., vii) says, speaking of laymen who have an opportunity to administer baptism: "He will be guilty of the loss of a soul, if he neglects to confer what he freely can," St. Jerome (Adv. Lucif., ix): "In case of necessity, we know that it is also allowable for a layman [to baptize]; for as a person receives, so may he give," The Fourth Council of the Lateran (cap. Firmiter) decrees: "The Sacrament of Baptism . . . no matter by whom conferred is available to salvation," St. Isidore of Seville (can. Romanus de cons., iv) declares: "The Spirit of God administers the grace of baptism, although it be a pagan who does the baptizing," Pope Nicholas I teaches the Bulgarians (Resp, 104) that baptism by a Jew or a pagan is valid.
Owing to the fact that women are barred from enjoying any species of ecclesiastical jurisdiction, the question necessarily arose concerning their ability to bestow valid baptism. Tertullian (De Bapt., xvii) strongly opposes the administration of this sacrament by women, but he does not declare it void. In like manner, St. Epiphanius (Hær., lxxix) says of females: "Not even the power of baptizing has been granted to them", but he is speaking of solemn baptism, which is a function of the priesthood. Similar expressions may be found in the writings of other Fathers, but only when they are opposing the grotesque doctrine of some heretics, like the Marcionites, Pepuzians, and Cataphrygians, who wished to make Christian priestesses of women. The authoritative decision of the Church, however, is plain. Pope Urban II (c. Super quibus, xxx, 4) writes, "It is true baptism if a woman in case of necessity baptizes a child in the name of the Trinity." The Florentine decree for the Armenians says explicitly: "In case of necessity, not only a priest or a deacon, but even a layman or woman, nay even a pagan or heretic may confer baptism."
The main reason for this extension of power as to the administration of baptism is of course that the Church has understood from the beginning that this was the will of Christ. St. Thomas (III:62:3) says that owing to the absolute necessity of baptism for the salvation of souls, it is in accordance with the mercy of God, who wishes all to be saved, that the means of obtaining this sacrament should be put, as far as possible, within the reach of all; and as for that reason the matter of the sacrament was made of common water, which can most easily be had, so in like manner it was only proper that every man should be made its minister. Finally, it is to be noted that, by the law of the Church, the person administering baptism, even in cases of necessity, contracts a spiritual relationship with the child and its parents. This relationship constitutes an impediment that would make a subsequent marriage with any of them null and void unless a dispensation were obtained beforehand.
I hope the above clears up any misconceptions you may have held about "the long-held Catholic tradition" regarding baptism.
It does-
but it’s still quoting Catholic Law.
I think we’ve gone as close together on this issue as we will get for now.
Good talking!