I hope this brings a little explanation

1662 Prayer Book on the private baptism of infants

"The Minister of every parish shall warn the people that without great cause and necessity they procure not their children to be baptized at home in their houses. But when need shall compel them so to do, then Baptism shall be administered on this fashion:
First, let the Minister of the Parish (or, in his absence, any other lawful Minister that can be procured) with them that are present, call upon God, and say the Lord's Prayer, and so many of the Collects appointed to be said before in the form of Publick Baptism, as the time and present exigence will suffer. And then, the Child being named by some one that is present, the Minister shall pour Water upon it, saying these words;
I baptize thee in the Name of the Father, and of the Son, and of the Holy Ghost. Amen.

Then, all kneeling down, the Minister shall give thanks unto God, and say,
E yield thee hearty thanks, most merciful Father, that it hath pleased thee to regenerate this Infant with thy Holy Spirit, to receive him for thine own Child by adoption, and to incorporate him into thy holy Church. And we humbly beseech thee to grant, that as he is now made partaker of the death of thy Son, so he may be also of his resurrection; and that finally, with the residue of thy Saints, he may inherit thine everlasting kingdom; through the same thy Son Jesus Christ our Lord. Amen.

And let them not doubt, but that the Child so baptized is lawfully and sufficiently baptized, and ought not to be baptized again. Yet nevertheless, if the Child, which is after this sort baptized, do afterward live, it is expedient that it be brought into the Church, to the intent that, if the Minister of the same Parish did himself baptize that Child, the Congregation may be certified of the true form of Baptism, by him privately before used: In which case he shall say thus, CERTIFY you, that according to the due and prescribed Order of the Church, at such a time, and at such a place, before divers witnesses I baptized this Child.

But if the Child were baptized by any other lawful Minister, then the Minister of the Parish, where the Child was born or christened, shall examine and try whether the Child be lawfully baptized, or no. In which case, if those that bring any Child to the Church do answer, that the same Child is already baptized, then shall the Minister examine them further, saying,
Y whom was this Child baptized?
Who was present when this Child was baptized?
Because some things essential to this Sacrament may happen to be omitted through fear or haste, in such times of extremity; therefore I demand further of you,
With what matter was this Child baptized?
With what words was this Child baptized?

And if the Minister shall find by the answers of such as bring the Child, that all things were done as they ought to be; then shall not he christen the Child again, but shall receive him as one of the flock of true Christian people, saying thus, CERTIFY you, that in this case all is well done, and according unto due order, concerning the baptizing of this Child; who being born in original sin, and in the wrath of God, is now, by the laver of Regeneration in Baptism, received into the number of the children of God, and heirs of everlasting life: for our Lord Jesus Christ doth not deny his grace and mercy unto such Infants, but most lovingly doth call them unto him, as the holy Gospel doth witness to our comfort on this wise."

As I understand there are two aspects to the normal baptismal service - firstly, put simply, the bringing of the child to Christ. Secondly the entrance of the child into the church - the latter being of necessity in the church building. The bringing of the child to the church I believe is why two records of baptism being given; there being no way other way to make an official record.
the coming