Sasha,
My understanding on Christian baptism is basically informed by three Scriptures.
Matthew 28:19, 20 shows me who it is that can baptize someone. I think it is saying that anyone who believes that commission applies to them is qualified to baptize other people.
Acts 2 shows me how Peter applied Jesus' words at Matthew 28:19, 20, and he did not use formalized words like, "In the name of the Father and the Son and the Holy Ghost . . ." It seems to have been the baptizee's perspective, their heart attitude, that was most important, not a specific rote and ritualistic invocation by the baptizer.
1 Peter 3:21 shows me what the Bible think baptism means. I can find no Scriptural reason to forbid someone from getting baptized again if they feel the need to do so.
I hope that helps.
Respectfully,
AuldSoul