Yes, you are considered Christian by both the Catholic Church and the Lutheran Church, but you would be further considered (by both) to be a
Lutheran Christian.
But they did not participate in the exact same
rite. It’s not the
priest that determines “Catholicity,” but the
participants and the
rite. In the Cattholic
Rite of Christian Baptism for Infants, the parents and sponsors are asked to affirm the Catholic Faith on behalf of the subject, including the question,
It is this affirmation, made within the understanding of the Catholic Church’s interpretation of “the holy Catholic Church” that separates Catholic Baptism from protestant Baptism. Both protestant and Catholic Baptism are the same valid Sacrament, but only one is truly Catholic, because only one expresses fidelity to the Catholic Church.