First and foremost, it is the initiatory rite into the Covenant.
I understand this to be the position of those who embrace Covenant Theology. Presbyterians (for example) baptize believers along with their children which places them under the "Covenant." They will not, however, baptize the infant of unbelievers or parents outside of their congregation.
For Baptists holding strictly to "Believer's Baptism," Baptism identifies you as one who has been placed IN CHRIST and RAISED TO WALK IN NEWNESS OF LIFE. It identifies you as a "Christ-Follower" and as a fellow believer. For this reason, I believe the congregation (or representative members thereof) should be present.
No one is saved by baptism, but no one can be considered a Christian without it.
I would agree with this statement. There may be a time period between which a new believer desires to be baptized and when such actually takes place and I have no problem considering them a fellow believer. If someone has not been baptized and REFUSES to do so, I would not consider them a "Christian" as you have stated.
Where was the congregation when Philip baptized the Ethiopian eunuch?
I was wondering whether this would come up and I was thinking about it when making my initial response!
The answer is that Philip is often referred to as "Philip the Evangelist" and the role of such is similar to the Apostles and whatever you would call a "Missionary" during that time (Paul was a "Missionary" and an "Apostle" who was charged with planting Churches and ordaining elders. Philip was likely sent out under the authority of his Church although it seems he had recently been placed in the office of a Deacon in the prior chapter but I digress.
He evidently had the authority to baptize and did so with the intent of planting and establishing churches. The Ethiopian eunuch had an "Entourage" with him who witnessed his baptism. Most likely were not saved but they were witness to his conversion and baptism nonetheless. He would be the beginning of a new congregation the kingdom of Ethiopia. Although little else is recorded, there was likely some Apostle or properly trained elder who followed up and established the congregation in which the Ethiopian Eunuch would be a part.
What about those who are incarcerated who come to faith? Should they or should they not be baptized? Under what authority should a baptism be administered? I believe it should be administered by an authorized representative of a Local Church congregation in the community - perhaps the congregation to which a Prison Chaplain belongs? There are prison chaplains who are on the State Payroll and I don't think they should be the ones baptizing. There are usually "Volunteer Chaplains" who are sent and supported by Churches which should administer the baptism. It may not be feasible for this person to be a member of this congregation but the congregation should take ownership of this individual and their continued spiritual growth. Perhaps some could and should wait until they are released from prison and be baptized by a Church congregation but this may not be a possibility for someone who is serving an extended or life sentence. In such a case, I would say that a prison baptism would be appropriate under the guidelines I have already stated.