If I might weigh in, thought not yet understanding the full context of your questions and this discussion…so I may have it wrong, but it seems like the overarching issue is in what way Baptism makes someone Catholic in the sense of being part of the Catholic Church.
I would recommend looking into the various Church statements about Salvation “Outside” the Catholic Church. These give context to both how salvation is given us (“Baptism now saves you”), and yet if/how salvation is possible for those not in full, open, visible communion with the Catholic Church. I was just yesterday researching this for some friends who were hearing people claim that the Church condemned all non-Catholics. Anyway, I find this topic very relevant to the topic of baptism and what constitutes some connection with the Catholic Church.
For reference: CCC818-819, 846-848; Lumen Gentium; Gaudium et Spes:22; Ad Gentes, no. 7; Dominus Iesus 6:22. These are referenced in the various links below.
Here on Catholic Answers:
Salvation Outside the Church
If there is no salvation outside the Church, then what does Christ’s death do for those who die without entering the Church?
Does “no salvation outside the Church” include non-Catholic Christians?
What “No Salvation Outside the Church” Means
What’s the correct understanding of “no salvation outside the Church”?
Forum threads:
Struggling with salvation outside the Catholic Church
Salvation Outside the Church?
The Catholic Church is necessary for salvation.
Extra Ecclesiam Nulla Salus (There is no salvation outside the Church)
Wikipedia (pretty good article):
Extra Ecclesiam Nulla Salus
To more directly address the baptism question, I think it’s neither 1 nor 2.
The Catholic Church is the Family (and Kingdom) of God. It contains all who have been reborn into the family of God through baptism (whether actual, by blood, or of desire, even implicit desire). It consists of all who have come out of the disunity of our race under Adam and into adoption by God through Christ, via an accepting response to God’s love in there lives.
Why do we call Pope Francis “Holy Father,” “Pope” (“Papa”)? He is a father to us in this family of God. In this sense, we can see that Pope Francis is indeed a father to more than just self-identifying Catholics in communion with him (for us he is a shepherd that we follow, as well, the first among the apostles on earth). He is a father to all who seek union with God. This is why, I think, so many non-Catholics are so interested in papal elections; there is a subconscious, spiritual yearning and recognition, motivated by the Spirit, that they may never realize but may nonetheless be very real.
Baptism is the formal, knowable sign of entry into the Everlasting Covenant of Christ. It can never be revoked, though we can turn our backs on it. This is because it operates like birth: once born (or conceived), you can never be un-born. You can die, you can even commit suicide, but you cannot undo what has been done.
Thus it is with baptism into the Catholic Church. Once baptized, always baptized. And yes, as I believe you have mentioned, it is the Spirit that does it, by Christ’s High Priestly power–ultimately the sincerity of the parents and Godparents and the state of the minister of the Sacrament are unimportant beyond the proper invocation of the Trinity and the intent to do what the Christ intends with the sacrament (which need not be understood by the minister; intent and assent only is needed, not understanding).
I have sometimes told non-Catholic friends that they are more Catholic than they know, simply by virtue of their baptism. For by that rebirth, our Church considers them official members of the family, however separated they may still be from full communion.
I hope this perspective might help the discussion.