Perhaps I can start with the simple fact that there is no example in the Bible of anyone being baptised into the name of the Father, the Son, and the Holy Spirit twice. And I think we should be very careful with any teaching or idea that has no command, promise, or example in God's Word. Our Lord's promise through Baptism is still valid, even if we wander off and by God's grace are brought back, for God does not lie, and we can rejoice in and take comfort in that!
When Scriptures say: "One Lord, one faith, one baptism", we can understand from this that this not only means "one kind of baptism" but also "a (singular) baptism in God's name".
Baptism is not something we do for ourselves or for God, but something God does for us — it's His gift to us. And the further away we get from understanding it as a gift, the less we know what Baptism is.
It can be good to read everything the Bible says about Baptism because the language the Holy Spirit uses to explain Baptism is very powerful. The Bible talks about Baptism in connection to faith, salvation, forgiveness of sins, a birth from above, the gift of the Holy Spirit, entering the kingdom of God, being baptised into Jesus’ death, a good conscience, a circumcision of Christ, a washing of regeneration and renewal of the Holy Spirit, a baptism into Christ, putting on Christ etc. So, in other words, a baptism into the triune name is not a symbol of being baptised in God's name, but it is being baptised in God's name, and that is God's work, gift, and promise to you. You are not baptised for God's benefit, but God has you baptised for your own benefit, to comfort you with His promise in a tangible way.
Peter sums all of this up wonderfully in 1 Peter 3:21: "Baptism, which corresponds to [the flood], now saves you, not as a removal of dirt from the body but as an appeal to God for a good conscience, through the resurrection of Jesus Christ."
Hope this helps! God bless!