James,
I hesitated to answer as I wasn't sure I had time to give it attention to help you get a handle on the way we understand what baptism signifies and to whom it is to be administered.
We, like the London Baptist Confession of Faith, believe that all became guilty in Adam in the Fall and were subject to the wrath and curse of God. We are all born not only guilty but corrupt in our natures so that all sin proceeds form that corruption and that we are justly deserving God's wrath.
We also both confess that God made a second Covenant, the Covenant of Grace with Christ and, in Him, all the Elect. Only those in Christ are Elect.
The fundamental difference in our theology is that the Baptist under their confession do not consider any administration of God's Covenants to be the Covenant of Grace until the New Covenant.
When a Baptist looks at the Old Covenant he sees a mixture with a promised Covenant of Grace that will be with Christ and all the Elect, but he sees primarily an administration of the elect and the reprobate participating in visible Covenants (Noahic, Abrahamic, Mosaic, Davidic) that exist to preserve an elect Seed for the coming of Christ at which time the Covenant of Grace is established in the New Covenant with Christ. They reason that if the mediators (OT Priests, Kings, Prophets) and the participants are imperfect then they cannot properly be considered the Covenant of Grace because the Covenant is made with Christ and, in Him, the Elect. That Covenant is "perfect" and there is no "imixture" of the bad king or reprobate member but only the Elect.
Presbyterians and the Reformed do not think this way. We see a Covenant of Grace under various administrations whose ultimate substance and fulfillment is Christ. We do not deny the imperfection of the mediators but we see the various signs and sacraments to unite all who visibly participate to and outward administration of the one Covenant of Grace. This is why the author of Hebrews can point us current believers to those in the wilderness and point out how they "rejected the Gospel" and fell away due to a lack of faith. We do not fundamentally see the sign of circumcision as physical sign only, but as a "sacrament" that has both visible and spiritual realities attached. Not everyone who was circumcised was elect but circumcision signified a circumcision of the heart (something only God could produce) and the Holy Spirit sealed that reality to those who truly drew near in faith.
Thus, when a Baptist reads of Isaac (as an infant) being circumcised he sees only a mixed Covenant (not really the Covenant of Grace) where Isaac receives it as a sign of things to come and as a way to preserve a people who will bring forth the Mediator who will inaugurate the Covenant of Grace into the future.
When a Presbyterian reads of Isaac's circumcision, he reads of a sacrament in which the visible sign (circumcision) signifies faith in the God who will bring forth the Promise of salvation. Isaac does not possess the *reality* of that sign merely by receiving it, but it belongs to God to bring that about by His sovereignty according to the purposes of election. That Isaac possesses faith doesn't owe to his circumcision but the sign does confirm to him a seal since God has promised to save him and so it operates on that level - he has faith and the sign sifnifies faith and salvation and so Isaac possesses both the sign and the seal.
When Isaac circumcised Esau and Jacob, he was not God and he obeyed the commandment to circumcise his children not as a bare sign of some physical future reality but in hope and on the promises of God. That Esau possessed a sign that signified faith does not nullify the fact that the sacrament of circumcision points to a reality that God promises salvation to all who believe. Esau was reprobate and, by his own actions and decisions, spurned the promise of God in his flesh and was justly condemned. Jacob, according to the election of God, was given a new heart and, for him, his circumcision was a seal of God's Covenant of Grace as well as a sign.
Fast forward to Christ and His death and resurrection. He commands to make disciples, baptizing them and teaching them to obey everything He has commanded.
The Baptist reads that command in light of his Covenant commitments.
Ah, he reads, now we have the reality of the Covenant of Grace and, he infers that since the New Covenant is perfect and only has the Elect within that we ought to only baptize the elect.
Now, at this point, he ought to be stumped because he doesn't know who the elect are.
But he is not deterred.
He moves on from this point and reasons: "Ah ha! I see in the NT that those who profess faith in Christ are those baptized and so I will only baptize those who profess Christ because, after all, the NT is perfect and those who are regenerate profess faith in Christ."
A disciple to the Baptist looks something like this. A disciple is someone who has been taught the tings of Christ and has been brought from death to life and is convinced he is a follower of Christ. He confesses his faith in Christ to the Church and the Church says: "Ah, we have a professor in Christ. We think he is therefore elect and regenerate and we are now going to baptize him on the basis that we think his profession is genuine. By doing so, we are upholding the "perfection" of the New Covenant that consists only of Christ and His elect."
The "speaker" in baptism in the Baptist schema is the believer. Because his profession is thought to be genuine then his baptism is valid. If at the time of baptism his profession was invalid then he was never really baptized. Thus, as a believer goes through the twists and turns of life he may later be assailed in his faith and conclude he never really believed. What is he to do? He is to profess faith NOW with a genuine profession and the Church will now baptize him for the first time. I say first time because it may be the second, third, fourth, ... time he has gotten wet but he was never really baptized. Why? Because only the elect are every really baptized and a person with a false profession is never baptized in Baptist thinking.
So, how does a Presbyterian think?
Well, Biblically. I say that with a smile even though I believe it.
We believe that Baptism is not the person's speech but God's speech. It is God Who makes promises in baptism.
If you read Matt 28:18-20, Christ commands the making of disciples by baptizing and teaching them everything He has commanded.
This completely blows a Baptist mind but we believe that a person is made a disciple is first by baptizing them and then teaching them. We baptize in order to teach.
We don't believe that the word disciple means elect. We believe that a person is made a disciple in the New Covenant to teach them the things of Christ and it is up to the work of the Holy Spirit to convert the hearts of any disciple. We believe that the Preaching of the Word and the administration of the Sacraments are the means the Holy Spirit uses in the lives of disciples to convert them.
Now, notice I talked about disciples first because we believe this about adults. An adult who professes faith in Christ is not presumed to necessarily be elect in the eyes of the Church. We don't have eyes to see that, nor is it our vocation to know this. We baptize adults who repent of their sins and turn in faith to Christ. Baptism is a sign that separates them from the world and visibly joins them to the visible Church. They are now disciples. Baptism does not, by its mere administration, confer salvation. It signifies salvation and a promise of salvation from God by the announcement of the minister.
We believe that an adult can have a false profession and fall away from the faith. We even believe that someone could have a false profession at baptism and, through years of preaching and the Sacraments, suddenly one day come to a saving knowledge where the Holy Spirit converts that individual. Every time I preach I remind people, as the author of Hebrews does, that Today is the day of salvation. If Today, you hear His voice, harden not your hearts.
If that person believes that word Today then what of his baptism years earlier? Well, remember what I wrote. Baptism is not my speech, it is God's Promise. Thus, the person with first-time genuine faith can look to the Promise of God in his baptism years earlier and know that God's promises are yes and amen. The Holy Spirit has now sealed and made real what the sign signified. The Sacrament of Baptism thus ties together sign and seal in a way where the person can look to his physical baptism and know with the Spirit that he is sealed with God's Promise. His baptism never did rest on the validity of his profession at the time of its administration but always on the promise of God.
So now what of our infant children? Well, I don't know if you connected the dots yet, but we consider them disciples. We baptize them as disciples in order to teach them everything God has commanded. We don't baptize them because they are elect but because, as the children of those in Covenant relationship with God, God has commanded the discipling of the children of believers. He has always commanded that His disciples disciple their children as members of the Covenant and the Covenant of Grace is now perfectly inaugurated and its Mediator is now in heaven as our Prophet, Priest, and King but it hasn't changed the basic rule of faith for families in the Covenant.
Our children are not guaranteed election or guaranteed faith. We train them in the fear and admonition of the Lord and, if they come to faith, they are like adult professors. The promise of their baptism is yes and amen.
If I was then going to summarize my answer to your questions, then, like Bruce, we do not know the status of infants as elect or reprobate. We are not commanded to doubt the salvation of our children who die as infants or small children. We do not believe that they are saved by the administration of baptism but, like any other sinner, only if they are united to Christ. The work of the Holy Spirit to choose and save anyone is inscrutable and so we don't presume to have the mind of God but look only to the Word where we are given confidence that those who are in Covenant with God do have a reason to trust that God has saved their children.
The reason we baptize them has nothing to do with our belief that baptism confers salvation by its administration but solely on the basis that, because thy are our children, they are in the Covenant and the sign of baptism belongs to them with all the duties for raising them in the fear and admonition of the Lord that this implies.