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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.