as far as the 16th century nonsense about baptism being for a public testimony, the baptism of the eunuch by Philip in Acts 8:26-40 blows this heresy right out of the water.
who did the eunuch give public testimony to when it was only he and Philip that went into the water? the answer is NO ONE. obviously Philip explained to him that baptism is for the remission of sins, receiving the Holy Spirit and being placed “into Christ”, that is why the eunuch was anxious to be baptized! if Philip told him baptism was for a public testimony, it was a waste of time because THERE WAS NO PUBLIC TO WITNESS IT!
You weren't there so obviously, your conjecture is meaningless...
They were not on an isolated two-track in the wilderness...They were on the main highway to Ethiopia...Could have been people all up and down the highway...
Besides, it would seem that one public witness would be plenty...