Hello Biker, the Didache was supposedly written by some of the Apostles, however, we are not sure of its exact authorship. Its greatest value to us is the look that it gives us into the faith and practice of the 1st Century Church (since the one thing that we do know is that it was written in the late 1st Century).
As for baptizing in the name of God, His ~singular~ "name" is, "
the Father and the Son and the Holy Spirit"
, according to the Lord Jesus anyway. Concerning this, I believe that we can be certain of two things,
1. that the Lord Jesus made no mistake concerning the baptismal formula that He gave us to use just prior to His Ascension (in Matthew 28:19) and,
2. that the Apostle Peter made no mistake when He baptized in the name of the Lord Jesus alone in Acts. The reasons that have given for this (for Peter's choice to not follow the Lord's baptismal formula in the Book of Acts) is nothing more than conjecture.
However, one 'very' important thing that St. Peter did for us (by baptizing in Jesus' name only in certain instances)
, was to show us/confirm for us that there is ~not~ a specific set of words that needs to be spoken during baptism, as if the words are some kind of necessary incantation that must be spoken for a baptism to be considered legitimate or proper.
That said, I suspect that the immediate reason that Peter chose to baptize in the Lord Jesus' name only was for the sake of those who were being baptized (because of their unique situation as 1st Century Jews)
, but again, anything like this that goes beyond points one and two above is simply a matter of conjecture.
We know from the Didache, and from the Ante-Nicene Fathers (as well from the later ECF)
that the church chose to follow Jesus' baptismal formula, not Peter's, and that they thought it was important, but not necessary*, to do so.
*(they could not have believed that it was "necessary" to follow the Lord Jesus' baptismal formula based upon the early chapters of Acts alone)
~Deut
Matthew 28
19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.