The term “Believers Baptism” is used in two ways. From what I remember a decade or two ago, it was just the idea that only professed, adult or teen, decided Christians should be Baptized.
More and more it seems to be connected to whether Baptism is an explicit step in justification as in one of your quotes. “Justification” from viewing the process as justification by faith (which comes solely by grace if adhering to election), and then sanctification by the Spirit of obedience which faith brings. So I have heard it more saying that “believer’s baptism” also implies that baptism is not part of justification but rather a show of obedience and/or a public showing and uniting with a particular church (denomination). I’m not a fan of this definition because it conflates the two. Someone who thinks only the decided should be baptized but sees it as part of the process of salvation can no longer express the first belief by saying “believers baptism” because they would be implying it is not part of the salvation process, but even there the two ideas are related.
So I agree this is a separate question. However, while the hyperlink is about whether Baptism should be Adult-only, there is a way in which the two are connected.
First about infant baptism. My understanding of the issue is that the first people to follow Christ (to become Christians) were only people who had decided to follow Christ, no one was (even tentatively) ‘born into it’. And then these new Christians would immediately be baptized in that faith (and yes John a bit a couple years prior). Some now say, “See. Only converted professed believers were being baptized, no babies.” But then others say, “Well that’s only because Christians didn’t exist yet so there weren’t Christian families where the whole family was Christians having babies. If there had been then of course they would baptize as soon as possible.” Hence the disagreement.
Thats not a full answer to what youre asking but it still reflects directly on your’s: Because if reading the Bible and you see that no one is being baptized except the converted who have decided and hence obviously have already professed.. then it might not (in such a person’s mind) take a separate instruction-verse telling you that’s how it works: 1. Profess. 2. Baptized.
Clearly, adults in Christ’s time presented with the opportunity to be Baptized would first decide and say whether they believe. For that matter just saying yes to baptism is doing so. I can see someone thinking that really is not the same as professing faith as a means of salvation and then just being Baptized as obedience.
One reason people can go there is by bringing in other verses about salvation by faith alone:
8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: 9 Not of works, lest any man should boast.
9 That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. 10 For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.
And others. Only by bringing in other verses would it be logical to see what was happening in Christ’s time and conclude that profession and baptism wasn’t the explicit justification process, unless one has a priori sentiments about faith vs activity, which I personally do, although the verses certainly help: personally I don’t think it should be seen as salvation making or breaking, shouldn’t divide the Body, and can be seen reasonably both ways. That said, why would anyone in either camp not do it? That makes it even less of a divider in my opinion.
Finally, it’s worth noting that they are also related in that if one believes it is explicitly part of the salvation process, then they would want to baptize their babies. Hence the correlation between the beliefs about Baptism.