Correct! As long as the baptism was done with water and using the correct words “I baptise you in the name of the Father, Son and Holy Spirit” and where the person doing the baptism has the correct intent of baptising the person, it is a valid baptism. This sacrament is one of two that do not need a priest. The other is Marriage, although marriages between a Catholic and another (Catholic or otherwise) do need to be performed by a priest or deacon to be valid in the eyes of the Church. But the priest does not confer the Sacrament, the couple confer it upon one another. So two non-Catholic Christians who marry outside the church have a sacramental marriage, even if they don’t know it.
And regarding your thoughts on baptism and the Mikvah, interesting thought. I think you are correct! I hadn’t thought of it before, but the parallels are certainly there. As has been mentioned however, the baptism is most closely aligned with the Bris, as both are to be performed when the child is an infant (according to Catholic teaching, Protestants differ on their view of infant baptism). Both instill an indelible sign that the person is a child of God (either part of his Chosen People for Jews, or actually an adopted member of God’s family for Catholics).
You will find that some Protestants object to infant baptism, insisting that one must first understand and accept Christ to be baptised. However, the Grace that one receives through baptism is a free gift from God, undeserved, as are all the Graces we receive. A child is worthy of such a gift only through God’s infinite Love and Mercy. Certainly no adult is more worthy of such a gift than would be an innocent child. It is then the responsibility of the parents and the God parents to ensure the child is taught the truth of Christ through the Church.
CARose