In their baptism, did they intend the same thing the Church intends?
I don’t know how to answer that question. We baptize new believers because Christ told us to and because it is a vital part of the faith of a new believer as they make a visible and conscious act of obedience to God. It is a celebration of the faith of a new believer and a physical display of the power of God to change our hearts. It is also a way to physically and outwardly represent what has happened inwardly and spiritually, not only in the heart and life of the new believer but also what has happened to all of us who follow Christ.
Baptism is normally done by a Pastor, however there are exceptions. The words spoken during baptism are something like this:
Pastor: Do you (baptism candidate) believe in Jesus Christ and is He your Lord and Savior?
Baptism Candidate: I do, or Yes, sometimes Yes, I believe in Jesus. In my current church the baptism candidate may also say or read a short testimony about how they came to faith in Christ and how it has effected their life.
The the Pastor will say, “Then upon your profession of faith in Christ I baptize you my brother/sister in the name of the Father, the Son, and the Holy Spirit”
The Pastor will then immerse the new believer and when they come up out of the water the pastor will say, “Buried with Him in Baptism, raised to walk in a newness of life”.
At that point the congregation will usually clap and maybe even cheer.
When several people are baptized, especially adults, it is a very moving service.
Immediately after their Baptism, did they preach against unity with the Catholic Church?
I spent the first 19 years of my life in the church I was baptized and I never remember the Catholic church even being mentioned. I’m not saying it wasn’t mentioned, but if it was it wasn’t memorable enough for me to remember.