And this authority is in Christ. Christ forgives our sins. Peter and the disciples acknowledged this. We do not have examples of Peter telling people, "I forgive your sins." He told them through belief in Jesus their sins were forgiven as noted in Acts 2 and 3.
This goes all the way back to Chapter 1, but to highlight one important passage in 13:10, Jesus says, "A person who has had a bath needs only to wash his feet; his whole body is clean. And you are clean, though not every one of you." Notice here that Jesus is talking about a cleansing--that is, a remission of sins--which occurs *after* a person has "had a bath", and which involves only "washing the feet" as opposed to the whole body--i.e., a remission that occurs after baptism.
Let's keep it in context. I agree the disciples' sins have been forgiven but the question is how?
From now on I am telling you before it comes to pass, so that when it does occur, you may believe that I am He. 20“Truly, truly, I say to you, he who receives whomever I send receives Me; and he who receives Me receives Him who sent Me.” John 13:19-20 NASB
We have the example of Cornelius also illustrating baptism does not forgive the sins of an individual.
43“Of Him all the prophets bear witness that through His name everyone who believes in Him receives forgiveness of sins.” 44While Peter was still speaking these words, the Holy Spirit fell upon all those who were listening to the message. 45All the circumcised believers who came with Peter were amazed, because the gift of the Holy Spirit had been poured out on the Gentiles also. 46For they were hearing them speaking with tongues and exalting God. Then Peter answered, 47“Surely no one can refuse the water for these to be baptized who have received the Holy Spirit just as we did, can he?” 48And he ordered them to be baptized in the name of Jesus Christ. Then they asked him to stay on for a few days. Acts 10:43-48 NASB
The whole Gospel of John is about faith/believe in Christ. In reference to John 6 Peter acknowledged it was about belief and not "eating and drinking" the flesh and blood which would be in violation of the OT prohibitions against eating the blood.
68Simon Peter answered Him, “Lord, to whom shall we go? You have words of eternal life. 69“We have believed and have come to know that You are the Holy One of God.” John 6:68-69 NASB
Regarding the passages from John and Acts where you highlight belief as a mechanism for the forgiveness of sins, yes, Jesus and the NT writers do emphasize the role of belief at times, but they never teach you are saved by "faith alone" as Martin Luther did (for more detailed discussion on what "faith" meant for Paul and 1st-century Judaism and why it didn't mean what Luther thought, I'd refer you to the work of E.P. Sanders). For example, in the Sermon on the Mount, when Jesus is teaching the Lord's Prayer, He says, "For if you forgive men when they sin against you, your heavenly Father will also forgive you. But if you do not forgive men their sins, your Father will not forgive your sins." (Matthew 5:14-15). When the Rich Young Man asked Jesus, "what good thing must I do to get eternal life?", Jesus answered, "If you want to enter life, keep the commandments," and then He got more specific about which commandments (Matthew 19:16f and parallel passages)--He did not say, "Believe alone." When He gives the Great Commission in Mark, He says, "Whoever believes and is baptized will be saved" (Mark 16:16); He does not say, "Whoever believes will be saved, don't worry about getting baptized." I could give more examples from the Lord--and His word is enough for me--but let's go on to the Apostles. James said, "As the body without the spirit is dead, so faith without deeds is dead." Luther wanted to take the Book of James out of the Bible because it did not fit his "faith alone" theology--no one taught "faith alone" before him. He based his "faith alone" theology on a misreading of Paul, but if we read the entirety of what Paul taught, he didn't teach what Luther taught. In the same passages where Luther thinks Paul is teaching "faith alone", we find Paul also saying things like, "We were buried with him through baptism into death in order that, just as Christ was raised from the dead through the glory of the Father, we too may live a new life" (Romans 6:4); and "if you live according to the sinful nature, you will die" (Romans 8:13); and "The only thing that counts is faith expressing itself through love." (Galatians 5:6) For Paul, as for Jesus (Matthew 7:12, 22:37-40, John 15:10, etc.), the key emphasis is on love (charity) working with faith, not faith alone. This is tied to what Paul says about the Holy Spirit, which IS love, because God is love (cf. 1 John 4:16). He talks about this in many places, but one key verse is Romans 5:5: "And hope does not disappoint us, because God has poured out His love into our hearts by the Holy Spirit, whom he has given us." The Holy Spirit, love, and faith are all tied together--along with hope, also mentioned in Romans 5:5 and in 1 Corinthians 13:13--because faith, hope, and love are all gifts--that is, graces--of the Holy Spirit. It is God's grace that saves us, in the form of the Holy Spirit poured into our hearts, which is at the same time a gift of faith, hope, and love. Hence in Ephesians 2:8, Paul says, "For it is by grace you have been saved, through faith--and this not from yourself, it is the gift of God--not by works, so that no one can boast." The faith that saves is not mere human belief, it is the gift of supernatural faith granted by the Holy Spirit--just as Jesus told Peter when Peter confessed He was the Christ, "Blessed are you, Simon son of Jonah, for this was not revealed to you by man, but by my Father in heaven." (Matthew 16:17)
Does acknowledging the role of faith, grace, or charity in salvation exclude the role of baptism? By no means! What does Paul say about baptism? He ties baptism to the gift of the Holy Spirit and to faith. I have already quoted Romans 6:4 as one example of this. Another is Galatians 3:26-27: "You are all sons of God through faith in Jesus Christ, for all of you who were baptized into Christ have clothed yourselves with Christ." He tells Titus, "He saved us through the washing of rebirth and renewal by the Holy Spirit" (Titus 3:5). Just as Peter says the water of Noah's Flood "symbolizes baptism that now saves you also" (1 Peter 3:21). And Jesus Himself says, "I tell you the truth, unless a man is born of water and the Spirit, he cannot enter the kingdom of God" (John 3:5)--and I have already quoted our Lord's words in Mark 16:16 as well.
What of Cornelius, then? You cite this as illustrating that baptism does not forgive the sins of an individual. But what does Peter say about it? Read on to Acts 11:15-16: "As I began to speak, the Holy Spirit came on them as he had come on us at the beginning. Then I remembered what the Lord had said, 'John baptized with water, but you will be baptized with the Holy Spirit.'" Peter himself identifies Cornelius' experience as baptism: baptism with the Holy Spirit--which we just saw Jesus talking about in John 3:5 while referring to water baptism in the same breath. And so what is the next thing Peter does as soon as Cornelius receives the baptism of the Holy Spirit? He immediately calls for water baptism: "Can anyone keep these people from being baptized with water? They have received the Holy Spirit just as we have. So he ordered that they be baptized in the name of Jesus Christ." (Acts 11:47-48) There are several things to observe here. One is that the Holy Spirit is the key factor in Christian baptism (as opposed to John the Baptist's baptism by water alone), and the Spirit can baptize by Himself without water and indeed without any human agency. However, far from excluding the need for water baptism, it calls for water baptism as the proper response by Peter (imitating the example set by Christ, who did not need to be baptized, but did so "to fulfill all righteousness"--Matthew 3:15). Water baptism normally accompanies Spirit baptism in the NT; Cornelius' case illustrates that God can also perform Spirit baptism distinct from water baptism if He so chooses, but that is His prerogative: it does not give believers a license to ignore Jesus' command to baptize, just as Peter insists that Cornelius must be baptized immediately after receiving the Holy Spirit. Catholic theology discusses Cornelius' case in relation to what is called baptism of desire, recognizing that God sometimes acts in exceptional circumstances by initiating Spirit baptism separately from water baptism, but also recognizing that He commanded us to perform water baptism at the time of conversion under normal circumstances. Water baptism thus "saves"--as Peter and Paul said in the passages quoted above--in the sense that it is normally the occasion of Spirit baptism, even if Spirit baptism sometimes happens first, as in Cornelius' case. Spirit baptism, water baptism, love, hope, and faith, are complementary, not contradictory.
Back to work for me--will check back on the thread as time allows. Meanwhile, God bless you.