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You are correct that Jehovah’s Witnesses reject infant baptism. We believe a person must be old enough to be a disciple. A disciple is a learner or taught one. Baptism is meaningless unless one is old enough to be a disciple. (Matt.28:19-20)
We also teach that full immersion is necessary to follow the manner in which our exemplar Jesus was baptized. In fact the Greek words translated baptize or baptism refer to immersing,dipping, or plunging under the water.
As far as the part of your question about a previous baptism in another religion being valid for Jehovah’s Witnesses, the answer is no.
Being baptized as one of Jehovah’s Witnesses is preceded by an extensive teaching program and a person will have to demonstrate that they have a basic understanding of bible teachings, live in harmony with them and give an affirmative response to the question “ Do you really want to be one of Jehovah’s Witnesses?”
Additionally there are over 100 other questions that are discussed to ascertain that a person meets the scriptural requirements for baptism.
Also as you point out in your question on the day of a new JWs baptism they answer yes to the question do you recognize that “your dedication and baptism identify you as one of Jehovah’s Witnesses in association with God’s spirit-directed organization.?”
Is baptism essential for salvation?
It is expected that a person who “comes to an accurate knowledge of Jehovah and the one He sent forth Jesus Christ” would be moved to dedicate their lives to Jehovah and symbolize it by being baptized.
Still we understand that baptism itself is not what saves. Salvation is possible only because God’s Son died a sacrificial death and was resurrected. Added to that is the requirement to maintain a good conscience before God by continuing to conform to His standards. Baptism only saves us if we remain faithful as witnesses of Jehovah. More info here
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The Jehovah’s Witnesses claim that if you were previously baptized into a different group, that baptism would be invalid. See para. 16 in this ‘Watchtower’ link, where they explained how they reasoned with a lady in her late 70s to get re-baptised in order to become a Jehovah’s Witness : https://www.jw.org/en/library/magazines/watchtower-study-march-2018/baptism-requirement-for-christians/
15, 16. (a) Is there a certain age at which one should get baptized? Explain. (b) Why must a Bible student get baptized as one of Jehovah’s Witnesses even if he was previously baptized in another religion? 15 There is no set age at which one should get baptized. Each student grows and matures at a different rate. Many get baptized at a young age, and they go on to be faithful to Jehovah. Other students see the importance of getting baptized after learning Bible truth when advanced in years—some being over 100 years of age! 16 One elderly Bible student asked her teacher if it was really necessary to get baptized again; she had previously been baptized in different religions over the years. The student and her teacher reviewed pertinent scriptures from the Bible. The student now understood the Bible’s requirement and got baptized soon thereafter. Although in her late 70’s, she did not feel that baptism was unnecessary for her. Yes, valid baptism is based on an accurate knowledge of Jehovah’s will. Hence, new disciples must get baptized even if they previously were baptized in another religion.—Read Acts 19:3-5.
If they claim that only their form of baptism is valid, then that is akin to saying, “Our form of baptism is essential, because no other can stand alongside it as equal, and we believe that Jesus commands water baptism if you are to become a Christian so you must submit to the Jehovah’s Witness form of baptism if you are to become a Christian.” Now, if I am reading more into that matter than is warranted, I hope some Jehovah’s Witnesses will put me right.I know that that exact wording is not found anywhere in any of their literature!This is just me paraphrasing what that article seems to be effectively saying – that only Jehovah’s Witness baptism is valid.
The rest of that article in the link says more about their views on baptism, which is only ever brief immersion under a body of water. Relevant to your question is paragraph 4 in that article, which states,
“A person who needlessly delays getting baptized endangers his prospects for everlasting life.”
This is based upon their understanding of 1 Peter 3:20-21, Mark 13:10 and Revelation 7:9-10. If I am not much mistaken, Jehovah’s Witnesses equate ‘everlasting life’ with salvation.
The two questions asked of their baptismal candidates show what is viewed as essential:
“On the basis of the sacrifice of Jesus Christ, have you repented of your sins and dedicated yourself to Jehovah to do his will? Do you understand that your dedication and baptism identify you as one of Jehovah's Witnesses in association with God's spirit-directed organization? Having answered yes to these questions, candidates are in a right heart condition to undergo Christian baptism."
Only the first question was given in the March 2018 link above, but both can be found in their 1st June 1985 ‘Watchtower’ magazine, page 30.