A few thoughts:
1) It seems that the 1st recorded instance of someone equating the sacraments of Circumcision & Baptism was the Apostle Paul:
Colossians 2:11-14 -- "In Him you were also circumcised with the circumcision made without hands, by putting off the body of the sins of the flesh, by the circumcision of Christ, buried with Him in baptism, in which you also were raised with Him through faith in the working of God, who raised Him from the dead. And you, being dead in your trespasses and the uncircumcision of your flesh, He has made alive together with Him, having forgiven you all trespasses, having wiped out the handwriting of requirements that was against us, which was contrary to us. And He has taken it out of the way, having nailed it to the cross."
2) The difference in terminology between Baptists & Reformed Christians, while often overlooked, points us to a major part of our differences. That is, Reformed Christians have "Sacraments" while Baptists have "Ordinances." An ordinance is a law, something one does b/c it is commanded. Sacraments are "signs & seals," promises by God to do something which is a myserty (Gk, "musterion"), that is, something which we understand only in part which, though formerly not understood (e.g. Circumcision as a testimony to Christ's redemption) is now revealed (e.g. Baptism as the NT fulfillment of the OT sacramental promises), and which points to a miraculous work of God the HS.
In other words, Baptists do not have sacraments. The Lord's Supper is a memorial. Period. We do it to remember Christ (note the nearly ubiquitous etching on Baptist communion tables), because He said so. Baptism is me obeying God's command to bear witness to what He & I have accomplished-- namely, my salvation. (After all, it points to my obedience to the Law by exercising faith. And never forget that to exercise faith is an act of obedience to the Law. [Interesting Master's thesis: "How does a Baptist view of baptism lead almost inevitably to Arminianism?"]) Baptism is therefore a sign, but not a seal. After all, the subtext of Baptist 'sacramentology' is not monergism, but cooperative grace.
By contrast, Reformed Christians do indeed have sacraments. We resist the popish & Eastern errors which make it magical, but we embrace the mysterious nature of God's sovereign promises, and revel in the fact that He has made promises to us & to our children, and look with eager expectation to the fulfillment of the sacramental promises in our children.
3) Yes, Reformed folk DO believe that our children are Christians. We believe that as those who share Abraham's faith (cf. Gal. ch. 3) we are sons & daughters of Abraham, citizens of the Commonwealth of Israel, and heirs of the Covenants of Promise (cf. Eph. ch.2). As such, we inherit the Sacrament of Circumcision as reinterpreted into the Sacrament of Baptism (which signify & seal the same things to Covenant children). Therefore, we eagerly look for the salvation of our children as promised by God to the elect of every age. HOWEVER, we do not presume regeneration. Even in the OT there was no presumption of regeneration (though perhaps this is edging into an anachronistic argument). Rather, every Covenant child is commanded to obey, and Covenant parents are commanded to teach their children "the way that they ought to go, that even when they are old they will not depart from it." Deut 6 is therefore a command to Covenant parents.
That said, there is the awful reality of Heb 6 & 10 hanging over all Covenant children who never exercise faith.
4) I do not believe that a Baptist can be Covenantal in the same sense that a Reformed Christian can be. The views of the relationship between the OT & NT are too divergent. Reformed folks believe in the fundamental CONTINUITY between the 2 dispensations of the Covenant of Grace. Baptists believe in a fundamental DISCONTINUITY between them. Indeed, it seems that it is almost impossible for a Baptist to adhere to a bi-covenantal structure of redemptive history (i.e. works in Adam, grace in Christ). Rather, a tri-covenantal structure seems necessary (Pre-fall, OT, NT). After all, each of the "three" has its own 'sacraments,' its own laws, its own telos, and never the thrain shall meet.
5) Consequently, there is no real correspondence (on a theological level) between a Baptist dedication and a Reformed baptism, at least as children are concerned. Indeed, due to the fact that Reformed folk have sacraments and Baptists have a sectarian rite, or ordinance, of baptism, I would say that the fundamental differences in ultimate theological significance of the two makes them quite distinct such that there is no correspondence.
I'm sure that both sides would agree, however, that we DO have the same 'sacraments.' We each view the other as inadvertantly doing the right thing, however inconsistently.
After all, does not the Reformed teaching that to neglect baptizing our children is a grave sin? And do not Baptists see the baptism of children as sadly errant?
We have mutually exclusive views on the relationship between the testaments, the sacraments, ecclesiology (arising from these differences), and the place of children in God's economy.
Sorry to be so long-winded, but that's my stab at addressing the OP.
Shalom,