I used to believe that when a person is Baptized in the Catholic church, in addition to Original Sin being removed from their soul, this was their official "joining the church" moment. Once a person is baptized they are now a Catholic and will be until they die. As I've become more traditional, my belief has shifted to the idea of Baptism simply being the process to remove Original Sin and the whole idea of "joining the church" being a modern addition. After all, we know that people who have been excommunicated, heretics and schismatics are no longer members of the church. In addition to that, protestant baptisms that use the proper form are considered valid, yet people who receive them are not immediately considered Catholic and cannot present themselves for Catholic sacraments.

Do I have it right or is there something I'm missing? Is Baptism only the removal of Original Sin or is it something more?

The adult who is baptized a Catholic, if he has the will to grow in the Catholic faith, becomes a member of the Church at his baptism. An adult who is correctly baptized a non-Catholic with no desire to grow in the Catholic faith, is cleansed of original sin but is still outside of the Church. For adults, faith is just as necessary as the sacrament itself in order to become a member of the Church.

Heretics and schismatics, if they ever were members, remain members of the Church, as Trent's catechism teaches: "Heretics and schismatics are excluded from the Church, because they have separated from her and belong to her only as deserters belong to the army from which they have deserted".

Please take note that although they belong to her only as deserters, the catechism says they still belong to her - this, in spite of their trying to get out of it by separating themselves from her via their heresy and schism. Once a Catholic, always a Catholic.      
 
Also note that the heretics and schismatics have separated themselves from the Church, the catechism does *not* teach that the Church has separated herself from heretics and schismatics - oh no, being members, they still have to answer to her, just the same as a deserter has to answer to the army - to which he still belongs.

I recommend always go the the basics for clarity. From the Catechism of St. Pius X:

1 Q: What is the sacrament of Baptism?
A: Baptism is a sacrament by which we are born again to the grace of God, and become Christians.
2 Q: What are the effects of the sacrament of Baptism?
A: The sacrament of Baptism confers first sanctifying grace by which original sin is washed away, as well as all actual sin if any such exists; it remits all punishment due on account of such sins; it imprints the character of a Christian; it makes us children of God, members of the Church, and heirs to Paradise, and enables us to receive the other sacraments.

Great idea! Hard to argue with the traditional Catechism.

"once a Catholic, always a Catholic"?

https://www.cathinfo.com/general-discussion/so-what%27s-wrong-with-the-term-%27once-a-catholic-always-a-catholic%27/

There's some (tiny bit of) truth to it, but it's misused by R&R types like Father Wathen and Stubborn.

In one sense, once someone has been baptized a Catholic and come into subjection to the Pope, that relationship remains.  That's why, for instance, those who were baptized Catholic, even if raised Protestant, cannot validly marry outside the Church ... because the Church retains this jurisdiction over them.

But it's patently false to say that a Catholic can never lose membership in the Church.  There's no Pope, no Church Father, no Doctor, and no approved Catholic theologians who has ever taught that one cannot leave the Church, lose membership in the Church, and go from inside the Church to outside the Church.  That is a novelty invented by Father Wathen to explain R&R ... and Stubborn has been pushing this nonsense around here for a long time.

Now, if one wanted to argue that the Baptismal character alone suffices to hold office in the Church, even without profession of the true faith and subjection to the Holy Father, then one could try to make that case.  But "once Catholic always Catholic" is ridiculous.

Satis Cognitum, On the Unity of the Church

Pope Leo XIII - 1896

https://www.papalencyclicals.net/leo13/l13satis.htm

5.

( . . . )

“See what you must beware of – see what you must avoid – see what you must dread. It happens that, as in the human body, some member may be cut off a hand, a finger, a foot. Does the soul follow the amputated member? As long as it was in the body, it lived; separated, it forfeits its life. So the Christian is a Catholic as long as he lives in the body: cut off from it he becomes a heretic – the life of the spirit follows not the amputated member” (S. Augustinus, Sermo cclxvii., n. 4).

(. . .)

9. The Church, founded on these principles and mindful of her office, has done nothing with greater zeal and endeavour than she has displayed in guarding the integrity of the faith. Hence she regarded as rebels and expelled from the ranks of her children all who held beliefs on any point of doctrine different from her own. The Arians, the Montanists, the Novatians, the Quartodecimans, the Eutychians, did not certainly reject all Catholic doctrine: they abandoned only a certain portion of it. Still who does not know that they were declared heretics and banished from the bosom of the Church? In like manner were condemned all authors of heretical tenets who followed them in subsequent ages. “There can be nothing more dangerous than those heretics who admit nearly the whole cycle of doctrine, and yet by one word, as with a drop of poison, infect the real and simple faith taught by our Lord and handed down by Apostolic tradition” (Auctor Tract. de Fide Orthodoxa contra Arianos).

The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium. Epiphanius, Augustine, Theodore :, drew up a long list of the heresies of their times. St. Augustine notes that other heresies may spring up, to a single one of which, should any one give his assent, he is by the very fact cut off from Catholic unity. “No one who merely disbelieves in all (these heresies) can for that reason regard himself as a Catholic or call himself one. For there may be or may arise some other heresies, which are not set out in this work of ours, and, if any one holds to one single one of these he is not a Catholic” (S. Augustinus, De Haeresibus, n. 88).

Heretics and schismatics, if they ever were members, remain members of the Church, as Trent's catechism teaches: "Heretics and schismatics are excluded from the Church, because they have separated from her and belong to her only as deserters belong to the army from which they have deserted".

Please take note that although they belong to her only as deserters, the catechism says they still belong to her - this, in spite of their trying to get out of it by separating themselves from her via their heresy and schism. Once a Catholic, always a Catholic.      
  
Also note that the heretics and schismatics have separated themselves from the Church, the catechism does *not* teach that the Church has separated herself from heretics and schismatics - oh no, being members, they still have to answer to her, just the same as a deserter has to answer to the army - to which he still belongs.

St. Robert Bellarmine. On the Church Militant (De Controversiis)

page 17

CHAPTER IV: On Heretics and Apostates

ALPHONSUS DE CASTRO teaches that heretics and baptized apostates are members and parts of the Church, even if they openly profess a false doctrine. Such an opinion is clearly false, and it can easily be refuted.

( . . . )


page 23

Thirdly, the argument is made that the Church can judge and punish heretics, therefore they are within it, “For what is it to me to judge those who are on the outside?” Besides, heretics retain the character of Baptism and priesthood, therefore they are Christians and priests.


I respond: Although heretics are not in the Church, nevertheless they ought to be; hence they pertain to her like sheep to the sheepfold when they roam outside the sheepfold. The Church can judge concerning those who are inside by that very fact, or who ought to be, just as a pastor really can judge and compel the sheep who wander outside of the sheepfold through the mountains to return to it. In the same way, a general can compel by force a deserter from the army who has fled across to the camp of the enemy to return or even to hang him. The Apostle, on the other hand, speaks on those who were never truly in the Church.

Now I speak to that which relates to the character. Heretics retain those indelible characters outside of the Church, just as lost sheep retain the branding in their back and deserters of the army military signs: but they are not in the Church for that reason because those characters do not suffice to constitute someone in the Church; otherwise the Church would also be in hell. St. Thomas Aquinas says that the damned are not members of Christ in either act or potency. Besides, the character does not properly unite a man with the head, rather it is a sign of the power of a certain union, and consequently, in hell they are recognized by that sign as men who were members of Christ. Nevertheless, that it does not unite them is clear since something that is invisible cannot unite outwardly, nor interiorly when it is not in act or when it is not an operative habit. For that reason St. Thomas places the first internal union in faith.

To understand this subject for what it is, all any faithful Catholic has to do is consider what they themselves would need to do in order to be forgiven for their sin of heresy / apostasy / schism. These three are sins, their nature makes them the worst of mortal sins, but that's what they are, mortal sins.

For us Catholics, going to confession (or receiving Extreme Unction in danger of death) are the only means in this world for the forgiveness of sins, these means are available only to Catholics. One who is not a Catholic cannot receive the Sacraments.

So God forbid, should you or I or any Catholic who has fallen into the above sins decide to repent, he can walk into the confessional just as surely as you or I and be absolved of those sins, just exactly the same as any other penitent Catholic, which is something those outside of the Church cannot do. 

Heresy, Apostasy and Schism are generally public sins. You cannot just go to confession and get private absolution. If you have joined a protestant sect or denied Our Lord Jesus Christ by being ritually circuмcised to become a Jєω. You are outside the Church. You must make a public profession of Faith and a public renunciation of your errors just as a heretic or Jєω who converts must do. Then you can get absolution.
The separation from the Church is public the rejoining the Church must also be public. If it is not public then Catholics must continue to avoid you. It would be a scandal for a priest to give Holy Communion to someone who had left the Church after only a private absolution.

I too have heard this before, I'm not even sure where I heard it,  but this is patently false. The Abjuration of Heresy is used primarily for adult converts seeking to enter the Church prior to their baptism or conditional baptism. Otherwise, it is not a requirement for the remission of those sins - unless required by the official censure itself, or the pope or bishop or confessor requires it.

To be forgiven, all the penitent has to do is the exact same thing all penitent Catholics do for the remission of their sins - go to confession.

If a person becomes a heretic not just privately doubting a truth of faith, but an actual heretic. Or an Apostate which means denying Our Lord Jesus Christ.
He gets absolved secretly in the confessional. Then shows up at the Communion Rail. The faithful are rightly scandalized that this heretic is given Communion. The priest can say nothing. He cannot say I gave him absolution from his sin. The priest cannot explain because it is all secret and under the seal of the confessional.

He is outside the Church and must re-enter it in the same public way that any heretic or infidel enters it.

Negative.

The penitent's sins are forgiven in the sacrament of penance. This is the only thing necessary for the forgiveness of sins. Public abjuration is not part of the sacrament, never has been. One of the reasons those sins are the worst of mortal sins is because they are typically accompanied with so much obstinacy and wilful, vehement opposing of the true faith, that men who are guilty of it seldom or ever return to the faith, but this is due to their own obstinacy.

When we go to confession, no matter what our sin(s), even the above sins, the priest *first* removes whatever censure the Church may have attached to our sins, *then* he forgives the sin:
".... May our Lord Jesus Christ absolve you: and I, by His authority, absolve you from every bond of excommunication, (suspension (for clerics)), and interdict, in so far as I am able and you are needful. Next, I absolve you from your sins, in the Name of the Father, and of the Son, and of the Holy Ghost. Amen."

The forgiveness of those sins are given in the sacrament alone, which is something those outside of the Church have no access to, the remission of those sins is in no way dependent upon a public abjuration.