Apparently Presbyterians departed from their father, John Calvin in terms of Baptism.
John Calvin claimed the Baptismal Regeneration, Infant Baptism, but I don't find Baptismal Regeneration in the Westminster Confession, but notice the Infant Baptism was still there. Infant Baptism means the Unbelievers Baptism.
However, though Presbyterians may accept the Baptism of Baptist churches, Baptist Churches cannot automatically accept the Baptism of Presbyterians because of the following:
Westminster Confession 1646
I. 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, or 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 Churchy until the end of the world.
II. The outward element to be used in the sacrament is water, wherewith the party is to be baptized in the name of the Father, and of the Son, and of the Holy Ghost, by a minister of the gospel, lawfully called thereunto.
III.
Dipping of the person into the water is not necessary; but baptism is rightly administered
by pouring or sprinkling water upon the person.
IV. Not only those that do actually profess faith in and obedience unto Christ, but also
the infants of one or both believing parents are to be baptized.
V. Although it be a great sin to contemn or neglect this ordinance, yet grace and
salvation are not so inseparably annexed unto it as
that no person can be regenerated or saved without it, or that all that are baptized are undoubtedly regenerated.
VI. The efficacy of baptism is not tied to that moment of time wherein it is administered; yet, notwithstanding, by the right use of this ordinance, the grace promised is not only offered, but really exhibited and conferred by the Holy Ghost, to such (whether of age or infants) as that grace belongeth unto, according to the counsel of God's own will, in his appointed time.
VII. The sacrament of Baptism is but once to be administered to any person.
http://www.reformed.org/documents/i....org/documents/westminster_conf_of_faith.html