The Reformed and Presbyterian understanding of covenant children, of course, is that they are to be included in the membership of the visible church. Our Westminster Confession of Faith (XXV.II) says the following:
"The visible church, which is also catholic or universal under the gospel (not confined to one nation, as before under the law), consists of all those throughout the world that profess the true religion; and of their children: and is the kingdom of the Lord Jesus Christ, the house and family of God, out of which there is no ordinary possibility of salvation" [emphasis added].
The children of believers, who have not yet professed faith (nor have denied it), are to be received as members with their parents. Baptism is to be administered at the time of admission into the church. Again, our Confession of Faith (XXVIII.I) says,
"Baptism is a sacrament of the new testament, ordained by Jesus Christ, not only for the solemn admission of the party baptized into the visible church; but also, to be unto him a sign and seal of the covenant of grace, of his ingrafting into Christ, of regeneration, of remission of sins, and of his giving up unto God, through Jesus Christ, to walk in newness of life. Which sacrament is, by Christ's own appointment, to be continued in his church until the end of the world.
Baptism is a sign and seal of God's covenant administered to all received into the visible church and the covenant community. As an Old Testament example, by way of analogy and biblical precedent, at the time of the institution of the sign and seal of circumcision, it was to be administered to all the males in Abraham's household. Our OPC Directory of Worship (IV.A.2) clearly states, "The baptism of infants is not to be unnecessarily delayed. Notice of intention to present a child for baptism must be given to the session by a parent who is a believer. The baptism of adults must await their public profession of faith in Christ."
Your specific question about the ages of 11 and 13 is significant. I believe that the reference to "infants" in the Confession of Faith (XXVIII.IV) and in the Directory for Worship (cited above) should be understood in terms of an age of a minor. Are these children to be considered as adults? As stated above, "The baptism of adults must await their public profession of faith in Christ." In some congregations some young people of that age are deemed by the respective session ready to take a communicants' class and to be received as communicant members, wherein a young person would be received as and treated as an adult in terms of membership status. This is understood to mean that there is evidence of a credible profession of faith and such necessary discernment to participate as a communicant member with all of the rights, privileges, and responsibilities belonging to such membership (which includes communing at the Lord's Supper; voting in congregational meetings, including the selection of church officers; etc.).
I emphasized the word "some" in the above paragraph. There is not a prescribed age in the Scriptures nor in our OPC Book of Church Order which prescribes the age at which it is appropriate to receive a young person as an adult professed believer. For example, in our own congregation, young people typically pass through an intensive study of the Shorter Catechism before being considered ready to profess faith. The session has adopted a plan of instruction that builds a study of the Shorter Catechism into the 10th grade Sunday School class. We have judged that that is typically an appropriate time for such an in depth study in the WSC to occur before a young person is ready to stand as an adult on their own profession in the congregation, and not simply on the basis of his or her believing parent(s).
In the specific cases of the young people you refer to in your inquiry, it is the responsibility of the parents and of your particular session to make a judgment about the spiritual discernment and the credibility of profession these children are able to demonstrate. It is good to pray for those men of the session and to submit to their judgment, unless it can be clearly demonstrated that they have erred in their judgment of these cases.
Your question is an important one in the life of a congregation. May God bless you and may he give his church much wisdom in such matters.
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