The confession states that the visible church consists of believers and their children. Historically, the question has been whether children are baptized and are at that point added to a particular church, or if they are members because they are born to believing parents and therefore may be baptized. If my memory is correct, Johannes Vos held the latter position.
Those more knowledgeable about the Scottish Presbyterians may have different insights, but baptizing childrean without adding them to the church would be aberrant in the Presbyterian understanding of the covenant.
The words "member" and "membership" now means something it once did not.
Until recently (19th century on), I think it has always been understood in confessional Churches that baptism brings one into the visible Church, and then participating in the Lord's Supper is a sign that the baptized person identifies with and assumes the covenant responsibilities of being a part of the visible Church.
But now we have added a modern concept of "church membership" which requires taking or signing a public membership vow or covenant to become a "formal member" of a particular church or congregation - something I would argue seeped into American Reformed communities from their close contact with the prevailing congregationalism of the land.
So I think we often are talking past one another when we use the words "member" and "membership."
This dissertation was helpful for me when I was studying this issue a few years ago:
https://www.calvin.edu/library/database/dissertations/Ahn_Sang_Hyuck.pdf. Dr. Hyuck puts Rutherford’s writings on the subject in context as he was responding to the Congregationalists.
Unlike the Presbyterians, the Congregationalists required four conditions (Articles) to be met before allowing a person to become a communicant member.
I believe the 2 most applicable parts of Rutherford's
The Due Right of Presbyteries are found in:
Part 1, CHAP. 5. SECT. 5. PROP. 3. QUEST. 6., pp.85-86 (I have updated the spelling):
3. Distinct. An explicit vocal Covenant whereby we bind ourselves to the first three Articles [see above] in a tacit way, by entering in a new relation to such a Pastor, and to such a Flock, we deny not, as if the thing were unlawful for we may swear to perform God's commandments, observing all things requisite in a lawful oath. 2. But
that such a covenant is required by divine institution, as the essential form of a Church and Church-membership, as though without this none were entered members of the visible Churches of the Apostles, nor can now be entered in Church-state, nor can have right unto the seals of the covenant, we utterly deny.
4. Distinct.
We grant a covenant in Baptism which is the seal of our entry unto the visible Church. 2. That it is requisite that such Heretics, Papists, Infidels, as be received as members of our visible Church, (from which Papists have fallen, having received baptism from us) do openly profess subjection to God, and his Church, in all the Ordinances of God. And that Infidels give a confession of their faith, before they be baptized. 3. Nor deny we that at the election of a Pastor, the Pastor and people tie themselves, by reciprocation of oaths, to each other, the one to fulfill faithfully the ministry that he hath received of the Lord; the other to submit to his ministry in the Lord, but these reciprocal oaths, make neither of them members of a visible Church, for they were that before these oaths were taken.
and in Part 2, CHAP. 4. SECT. 5., pp.185-186 (in this section Rutherford is responding to the Congregationalist position “Why we do not admit the Members of the Churches of Old England to the Seals of the Covenant”). I have again modernized the spelling:
Quest. I. Whether the Seals of the Covenant can be denied to professors of approved piety, because they are not members of a particular visible Church, in the New Testament.
Our Brethren deny any Church Communion, and the seals of the Covenant, Baptism, to the children of Believers, the Lord's Supper to believers themselves, who come to them from Old England, because they be not members of the particular Congregation to which they come, and because there is no visible, Church in the New Testament, but one particular Parish, and all who are without a particular Parish, are without the visible Church, and so are not capable of either Church censures, or the Seals of the Covenant, because they have right to the seals of the Covenant, but only this visible Church.
We hold all who profess faith in Christ, to be members of the visible Church, though they be not members of a visible Congregation, and that the seals of the Covenant should not be denied to them. And for more full clearing of the question, let these considerations be observed.
First, Dist.
All believers, as believers, in foro Dei, before God have right to the seals of the Covenant, these to whom the Covenant and body of the Charter belongeth, to these the seal belongeth, but in foro Ecclesiastico, and in an orderly Church-way, the seals are not to be conferred by the Church upon persons because they believe, but because they profess their believing: therefore the Apostles never baptized Pagans, but upon profession of their faith.
Second Dist. Faith in Christ truly giveth right to the seals of the Covenant, and in God's intention and decree, called voluntas beneplaciti, they belong only to the invisible Church, but the orderly way of the Churches giving the seals, is, because such a society is a professing or visible Church, and orderly giving of the seals according to God’s approving will, called, voluntas signi & revelata, belongeth to the visible Church….
We hold that those who are not members of a particular Congregation, may lawfully be admitted to the seals of the Covenant.
First, Because those to whom the promises are made, and profess the Covenant, these should be baptized. But men of approved piety are such, though they be not members of a particular Parish. The proposition is Peter’s argument, Act. 2. 38.
Secondly, Those who are not Members of a particular Church may be visible professors, and so members of a visible Church, Ergo, the seals of the Covenant belongeth to them.
Thirdly, The contrary opinion hath no warrant in God's Word.
Fourthly, The Apostles required no more of those whom they baptized, but profession of belief, as Act. 10. 47. Can any forbid water that these should not be baptized, who have received the Holy Ghost, as well as we? Act. 8. 37. If thou believest with all thy heart, thou mayest he baptized: no more is sought of the Jailor, Act. 16. 31. 34.
And there is, of course, AA Hodge's famous quote: " "A Church has no right to make anything a condition of membership which Christ has not made a condition of salvation." (from his commentary on The Westminster Confession of Faith, p.5)