The page gives your definition. I am look for the Church's definition of a public manifest formal heretic.
Pius XII didn't used the term sin, or the phrase "public manifest formal heresy." He used the term admissum, which is a general term that can be sin, crime, or offense. The reason he used that term specifically is because there are two schools of thought concerning what is required to be a member of the Church. One holds that the external bonds alone suffice, the other maintains the external bonds and interior faith are necessary to be a true member of the Church. Since sin destroys interior faith, if Pius XII had used the word sin - the sin of heresy severs a person from the Church by it nature - he would have been seen as teaching the latter opinion. To avoid the appearance of weighing in on this centuries old debate, he used the ambiguous term admissum.
But let's see what Pius XII says in the very next paragraph:
Pius XII: "24. Let every one then abhor sin (peccatum), which defiles the mystical members of our Redeemer; but if anyone unhappily falls and his obstinacy has not made him unworthy of communion with the faithful, let him be received with great love, and let eager charity see in him a weak member of Jesus Christ. For, as the Bishop of Hippo remarks, it is better "to be cured within the Church's community than to be cut off from its body as incurable members. ' (August., Serm., CXXXVII)."
If a person falls into sin without, yet is obstinacy has not made him unworthy of communion with the faithful (had retained the external bonds), he is to be received with greater love, for, as Augustine teaches, it is better "to be cured within the Church's community than to be cut off from its body as incurable members."
To gain insight into the mind of Pius XII, we should ask who St. Augustine was referring to in the quote he cited? Was he referring to someone who was only guilty of a moral offense, or was he referring to those who openly attacking the faith, and therefore have apparently fallen into the sin of heresy, yet remained in communion with the Church? It was the latter. In the quote Pius XII cited, the Bishop of Hippo is speaking of the Pelagians who had not yet separated from the Church and joined a Pelagian sect. Here is the full quote in context:
St. August., Sermin, CXXXVII: “But because he went off, having been found guilty and detested by the Church rather than corrected and subdued, I was afraid that it was perhaps he himself who was trying to disturb your faith, and for this reason I thought I should mention his name. But it makes no difference whether it is he or others who partake of his error. For there are more than we would expect. And where they are not refuted, they win over others to their sect, and they are becoming so numerous that I do not know where they will turn up. Yet we prefer that they be healed within the framework of the Church rather than cut off from its body like incurable members, at least if the very gravity of the situation permits this.”
So, when Pius XII said "if anyone unhappily falls and his obstinacy has not made him unworthy of communion with the faithful, let him be received with great love, and let eager charity see in him a weak member of Jesus Christ," he was including those who fall into errors against the faith, and who spread them openly. As long as they remain within the framework of the Church, they remain part of the Church's body. And remaining part of the body suffices for a person to hold office in the Church.
"The common and general meaning of the word 'admissum' is defined by Lewis & Short as a 'voluntary fault', and only in certain specific instances can it be understood to mean 'crime', when the particular context in which it is used supports that interpretation. Salza & Siscoe gratuitously interpret the term as used in Mystici Corporis to mean 'offense” as in 'crime' – a canonical delict or transgression of ecclesiastical law which, in the case of heresy by ecclesiastical authority incurs the penalty of excommunication latæ sententiæ. It is quite impossible, and in fact, contra rationem, for the word 'admissum' to be understood as meaning only the act of of heresy, schism, or apostasy suapte natura in virtue of its being a canonical delict, separates a man from the Church, but not that the manifest external sin per se by its very nature accomplishes that separation in the context that it is used in this passage of Mystici Corporis, because that would render the meaning of the passage unintelligible and entirely irrational.
"Salza & Siscoe go to great lengths to insist that in Mystici Corporis, the words 'admissa' and 'admissum' mean, 'crime(s)', and not 'sin(s)'; but when you examine the syntax of text very carefully, it makes no difference how you translate the terms. Read the Latin text very carefully – it says: «And thus not every fault (admissum: sin, fault, crime), even a gravely evil deed (scelus: an evil deed; a wicked, heinous, or impious action; a crime, sin, enormity) does such – as schism, heresy, and apostasy do – by their very nature separate a man from unity of the body the Church.» There it is: Others are separated from the Church by excommunication – «by the legitimate authority of the Church» for having committed excommunicatable penal offenses, i.e. crimes; as opposed to those who «miserably separate themselves from union with the body» of the Church by heresy, schism, or apostasy, which separate them not for their being crimes punishable by the authority of the Church, but because they are of the nature of sins opposed to the unity of the Church; which therefore, according to their nature (suapte natura) separate the perpetrator from the body of the Church. In Canon Law, it pertains to the nature of a crime per se that it is a penal violation – a violation of a law or precept that is of ecclesiasticasl character; and, if the transgression is public, and if there is added to the law or precept the penal cenure of excommunication, it results in the separation of the offender from the Church by means of the penalty of excommunication, incurred or inflicted by the authority of the Church. Pius XII teaches, (in conformity with the constant teaching of the universal magisterium), that heresy, schism and apostasy, are the sole exceptions, because, although they in fact happen to be crimes; heretics, schismatics, or apostates are not separated from the body of the Church 'by legitimate authority', i.e. because they committed crimes; but because these sins by their very nature are directly and per se opposed to the unity of the Church; and accordingly, schismatics, heretics and apostates have <<miserably separated themselves from the unity of the Body» of the Church (a Corporis compage semet ipsos misere separarunt). The reason why this is so (as is explained below) is because the specific nature of each of these sins, i.e. of heresy, schism, (and a fortiori apostasy), is such that they directly and per se separate one from the unity of the Church. (IIª-IIae q. 39 a. 1 ad 3) On the other hand, criminal acts considered under their formal aspect as crimes, i.e., according to the nature of crimes, do not directly and per se separate one from the Church; but according to the nature of crimes as such, it is only by means of juridical authority that the separation would take place, being that they are crimes carrying the penalty of excommunication."
Kramer, Paul. To deceive the elect: The catholic doctrine on the question of a heretical Pope . Kindle Edition.