That's really does not concern you. That is God's business. If you have preached the true Gospel and someone says that they have believed on the Lord Jesus Christ, that must be accepted in good faith. See Acts 8.The thin[g] we don't like is people getting water baptized and thinking they are saved.
That's a given and should not even be raised as an issue. No one here is saying that water saves anyone, so that is just a red herring.Water does not save anyone.
That's also a given, but it DOES NOT PRECLUDE water baptism. And therein lies your error. Water baptism is a commandment of Christ, not an option for you or anyone to dismiss on a whim.People need to be baptized in Holy Spirit.
Well let's see if that is really the case: 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. For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.The actual baptism is found in Romans 10:9, 10. It's not water, but faith or genuine trust.
As you can see, this passage has nothing to say about baptism, but much to say about (1) believing on the Lord Jesus Christ and (2) confessing with your mouth the Lord Jesus.
And where and when do people confess with their mouth their faith in the Lord Jesus Christ? It is at the time when they are publicly baptized by immersion! They give their Christian testimony and then they are baptized.