You say that Cornelius and his family were saved at the time when the Holy Spirit came upon them in Acts 10. But in Numbers 24:2, the Spirit of God also comes upon Balaam, yet there is every indication from other verses about Balaam that he remained devoted to divination both before and after that event (See 2 Peter 2:15, Revelation 2:14, and finally Joshua 13:22). Deuteronomy 23:5 makes it clear that God intervened to make Balaam bless the Israelites; the blessing was not motivated by any kind of true repentance on his part. My contention is that Cornelius and his family were not saved when the Spirit came upon them. They were saved by their faith during water baptism afterward.
You are correct, the Holy Spirit did come upon Balaam to move him to bless Israel.
When Balaam saw that it pleased the LORD to bless Israel, he did not go as at other times to seek omens but he set his face toward the wilderness. And Balaam lifted up his eyes and saw Israel camping tribe by tribe; and the Spirit of God came upon him. And he took up his discourse and said, “The oracle of Balaam the son of Beor, And the oracle of the man whose eye is opened; the oracle of him who hears the words of God . . .” (NASB) Numbers 24:1-4
You are also implying that this does not mean that Balaam was a Christian and you are correct. Jesus made this point while He was here on this earth. The Holy Spirit is active in the lives of non-Christians and Christians. His goal is to bring them to Christ. The Holy Spirit influences people. That is the message of Jesus’ words.
And He, when He comes, will convict the world concerning sin, and righteousness, and judgment; concerning sin, because they do not believe in Me; and concerning righteousness, because I go to the Father, and you no longer behold Me; and concerning judgment, because the ruler of this world has been judged. (NASB) John 16:8-11
When we come to Acts 10, we find Cornelius, who is described as a God fearing man (Acts 10:1-2), seeking to know God. God responds and moves the apostle Peter to go to Cornelius for the purpose of telling him about Jesus Christ Acts 10:34-43 records the words that Peter shared with him and his family.
. . . Jesus of Nazareth, how God anointed Him with the Holy Spirit and with power, and how He went about doing good, and healing all who were oppressed by the devil; for God was with Him . . . And they also put Him to death by hanging Him on a cross. God raised Him up on the third day, and granted that He should become visible . . . Of Him all the prophets bear witness that through His name everyone who believes in Him receives forgiveness of sins.” (NASB) Acts 10:38-40, 43
While Peter was sharing this truth, the Holy Spirit “fell upon” all those who heard.
While Peter was still speaking these words, the Holy Spirit fell upon all those who were listening to the message. And all the [Jewish] believers who had come with Peter were amazed, because the gift of the Holy Spirit had been poured out upon the Gentiles also. For they were hearing them speaking with tongues and exalting God. Then Peter answered, “Surely no one can refuse the water for these to be baptized who have received the Holy Spirit just as we did, can he?” (NASB) Acts 10:44-47
What happens here is not the same thing that happened to Balaam. Those who heard the message ended up speaking in tongues. But Balaam did not. The apostle was surprised and so were those who were with him. This display of the Holy Spirit indicated they had believed in Jesus.
How Does One Believe?
When does a person start to believe in Jesus? Belief does not come after a prayer or at a baptism. It happens in the mind, heart and soul. At the moment they believed, the Holy Spirit fell on them. Nothing else had to happen. It is the Holy Spirit who places us into the body of Christ (1 Cor. 12:13). He knows when someone believes.
Later we discover in Acts 15, the apostles held a conference in the city of Jerusalem to determine if God was saving the Gentiles, too, and not just the Jews. Listen to the apostle Peter.
And after there had been much debate, Peter stood up and said to them, “Brethren, you know that in the early days God made a choice among you, that by my mouth the Gentiles should hear the word of the gospel and believe. And God, who knows the heart, bore witness to them, giving them the Holy Spirit, just as He also did to us and He made no distinction between us and them, cleansing their hearts by faith.” (NASB) Acts 15:7-8
Peter’s point is that the proof they believed in Jesus and were saved (Acts 15:11) was the immediate display of the Holy Spirit “just as He also did to us.” God cleansed their hearts by faith. There is no mention of baptism or water in the entire passage.
Water baptism is a public demonstration of our submission and dependence on Jesus Christ for the forgiveness our sins. Water does not save us. Faith or our depending on Jesus to forgive our sins is what is required to save .