So infants can be baptized in any of those modes, and would you see the Adult baptism in same way that the infant one? Both now part of the NC, or is it different as the adult now has made a profession of their faith in Jesus now?
I would understand the difference between the baptized child and the baptized adult like the difference between Abraham and Isaac when they received the sign of circumcision. Abraham received it
after he believed God and was credited with righteousness, but Isaac received it
before. Ishmael also received it, and never believed.
When Abraham received the sign, he was clearly
already a believer. When Isaac received it, it was unclear as to whether he would follow in the footsteps of faith of his father. This did not affect the application of the sign. The sign was to set apart members of the believing community; children were included in this community until they proved by unbelief that they did not belong. The application of the sign was independent of God's sovereign elective purposes.
It must also be noted that not only was the sign of circumcision a sign of promised blessings, but it was also a call to a holy life. Recall how the Lord urged his people to be 'circumcised in heart'; that is, to truly live as you have been called.
I see baptism in much the same way. While there are no explicit mentions of infant baptism in the NT, the lack of a prohibition and the similarities between baptism and circumcision (both are signs and seals which point to Christ's work, one looks forward, the other backward) leads me to believe that it is acceptable to baptize infants as long as one parent is a believer. I could be wrong, but that's how I see it.
The sign of baptism recognizes what the Lord has done. In the case of the newly converted adult, it recognizes the fact that God has already washed them clean of sin, grafted them into his family, set them apart, and given them his Holy Spirit. For the covenant child, I see at as also a recognition of what God has done - he has placed that child into a believing household (independent of the child's choice) which should ensure that the child has the opportunity to hear the gospel and respond in faith. The baptism of a child is also a call on that child to live a holy life and respond in personal faith to the gospel offer, which is signified in their baptism.
As for their part in the NC, I will respond to that later.
Best regards,
Izaak