In the 17th century, three main reformed camps were the Presbyterians, the Congregationalists, and the Baptists. The Presbyterians believed in a national church. Congregationalists believed the church consists of people who had been saved out of the world (the nation) who then gather together in congregations. No one was considered a member of the church until they had made a credible profession of saving faith and was thus admitted to the Lord’s Table. Baptists were Congregationalists who rejected infant baptism.
Keach recognized a great inconsistency in the Congregationalists. In fact, it was specifically this inconsistency that led to a controvery in New England, resulting in the Synod of 1662. The Congregationalists were faced very practically with Keach’s dilemma. They could not answer it, so they abandoned a key Congregationalist tenet and turned back towards a Presbyterian view, which did not restrict membership to those who had been called out of the world. This is known as the Half-Way Covenant.
As for our Brethren, called Congregational, I cannot tell what they mean by contending for the Practice of Pædo-Baptism, nor do I well know what their Sentiments are about it: they agree (as I do understand) with us (and other Christians,) that Baptism is an initiating Rite or Ordinance; now if their Infants are in Covenant with themselves, and are made visible Church-Members by Baptism in Infancy, and until by actual Sins they violate their Right and Privilege, abide Members thereof.
(1.) Then I would know whether they have their Names in their Church-Book, or Register, as Members? And
(2dly,) Whether they ever Excommunicate (or bring under any Church Censure) such of their Children who fall into scandalous Sins, or actual Transgressions, or not?
(3dly,) If not, what kind of polluted Churches must thir’s be, who have not purged out such corrupt Members?
The truth is, I see not how Infant Baptism is consistent with any Church State, unless it be National; and no doubt, the first Contrivers or Founders of it, devised that way for the Progress of that they call the Christian Religion, and so opened a Door, that Christ shut, when he put an end to the National Church of the Jews.—Therefore I wonder at our strict Independants, considering their Notions, (knowing how their Principles differ from; and their Understanding or Knowledge of Gospel-Church Constitution exceeds others) for Baptism does not initiate into their Churches, it seems by their Practice; unless their Children, when baptized, were thereby made Members with them.
Keach, B. (1693). Sermon III. In The Ax Laid to the Root, Parts I & II (Vol. 2, p. 34). London: John Harris.
(Note that modern Presbyterians, in abandoning the national church model, have followed largely in the path of Congregationalism)