Acts 2:14-21 does not say they received Jesus as Lord and Saviour.
It says they asked Peter “Brethren, what shall we do?” (vs 37)
Peter answered them “Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit." (Vs 38)
Then in vs 41 it says "So those who received his word were baptized, and there were added that day about three thousand souls"
So what were they added to? The answer can only be added to the Church, to those being saved.
You say that it never says they were baptised in water. But then it never says they were baptised with the Holy Spirit either.
But what would baptism mean to the apostles Peter and to the 3,000?
To Peter it meant baptism with water. As Peter says in Acts 10:47-48
“Can any one forbid water for baptizing these people who have received the Holy Spirit just as we have?” And he commanded them to be baptized in the name of Jesus Christ."
Baptism "the name of Jesus Christ" is with water.
In Acts 2:38 Peter tells the 3,000 they must be baptised "in the name of Jesus Christ". That must therefore be baptism with water.
Again in Acts 8:35-38 says nothing about accepting Jesus as Lord and Saviour. But they were clearly baptised with water.
In both cases we see the gospel being preached, people believing and then bein baptised.
Jesus said "He who believes and is baptized will be saved" Mk 16:16, so we know that after baptism they had been saved.
The person who baptised the eunuch in Acts 8 was Philip; the same Philip who was baptising earlier in Samaria. Of this baptism (in Samaria) it was they had "only been baptized in the name of the Lord Jesus." Again baptism in the name of Jesus was with water.
But note that subsequent to being baptised with water Peter and John laid hands on them and prayed and then they received the Holy Spirit (vs 17). But this is not the same as receiving the Holy Spirit in baptism. Earlier (vs 16) it said that the Holy Spirit "had not yet fallen on any of them". That is the external work of the Spirit. (note the word I have emboldened)
Compare that to Acts 19 with Paul and the 12 men in Ephesus. They had only been baptised in with John's baptism. That was not sufficient. Paul told them to be baptised "in the name of the Lord Jesus" (vs 5). That's baptism with water for salvation.
After that Paul laid hands on them and "the Holy Spirit came on them; and they spoke with tongues and prophesied." Again the exterior work of the Spirit., and came upon them - the same as fallen on them.
Two different actions. Two different purposes.
The Jerusalem Bible
1 John 5:6 Jesus Christ who came by water and blood, not with water only, but with water and blood with the Spirit as another witness since the Spirit is truth 7 so that there are three witnesses, 8 the Spirit, the water, and the blood and all three of them agree
KJV
1 John 5:6 This is he that came by water and blood, even Jesus Christ; not by water only, but by water and blood. And it is the Spirit that beareth witness, because the Spirit is truth. 7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. 8 And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one.
Literal water has nothing to do with salvation as Jesus came by the foretold word of the Prophets, sent by the Father, and signified by the Holy Spirit that lighted upon Him during John's water baptism for the remission of sins, Matthew 3:11-17. Three that bear witness in earth, the Spirit, the water (word) and the blood. Three that bear record in heaven, the Father, the Word and the Holy Ghost.
The more I think about "born again" Nicodemus had no understanding as even many today would ask the same question "How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?"
Jesus goes on to tell him "Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit."
Ephesians 5:26 That he might sanctify and cleanse it with the washing of water by the word,
Paul understood it was not by water baptism (BTW baptism is not mentioned in John 3:5), but by the word of God that we are sanctified, which means set apart and made holy. We being the church of God have been made clean by His word as we are washed in the blood of the Lamb.
Titus 3:5 Not by works of righteousness which we have done, but according to his mercy he saved us, by the washing of regeneration, and renewing of the Holy Ghost; 6 Which he shed on us abundantly through Jesus Christ our Saviour; 7 That being justified by his grace, we should be made heirs according to the hope of eternal life.
1 Peter 1:3 Blessed be the God and Father of our Lord Jesus Christ, which according to his abundant mercy hath begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead,
Again Paul and also Peter gives us the understanding that being born again/born from above is not by actual water which would be a work of our own righteousness, but regeneration (new Spiritual life) that comes by God's mercy through His spoken word and Spirit (Holy Spirit).
Ephesians 2:8 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. 10 For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them.
Per the conversation Nicodemus and Jesus were discussing in John 3:9, 10 here are a few more OT verses that parallel being born again that Nicodemus should have known, but probably not emphasized as much since Jesus had to explain it to him.
1 Samuel 10:6 And the spirit of the LORD will come upon thee, and thou shalt prophesy with them, and shalt be turned into another man.
Job 15:14 What is man, that he should be clean? and he which is born of a woman, that he should be righteous?
Job 25: 4 How then can man be justified with God? or how can he be clean that is born of a woman?
Ezekiel 11:19 And I will give them one heart, and I will put a new spirit within you; and I will take the stony heart out of their flesh, and will give them an heart of flesh: 20 That they may walk in my statutes, and keep mine ordinances, and do them: and they shall be my people, and I will be their God.
There are around 1526 instances of living water in the Bible. Here are some verses on living water being the word. John 4:4-26; 7:37-39; 12:44-50; Ephesians 5:26; 1 John 5:5-8; Jeremiah 17:13; Zechariah 14:8, 9; Rev 21:6-8; Ezekiel 47:22