Good question. Here is the passage:
30 Then he brought them out and said, “Sirs, what must I do to be saved?” 31 And they said, “Believe in the Lord Jesus, and you will be saved, you and your household.” 32 And they spoke the word of the Lord to him and to all who were in his house. 33 And he took them the same hour of the night and washed their wounds; and he was baptized at once, he and all his family. 34 Then he brought them up into his house and set food before them. And he rejoiced along with his entire household that he had believed in God.
I would argue that it is very important WHO Jesus is and WHAT He did. Verse 32 is important. I believe this is where the Gospel was shared. The jailer didn't just believe in some guy named Jesus. He knew who He was and what He accomplished. Does this make sense?
For example. The LDS Jesus is much different than my Jesus. Same with the Jehovah Witness Jesus and the Unitarian Jesus. Similarily, the Arminian Jesus is different than the Jesus I worship.
Hopefully you can admit that not everyone who "Believes on the Lord Jesus will be saved" in that context.
I believe that words and names are imbued with meaning that is independent of what the person speaking them believes they mean. The classic example of this is when Isaac put his hands on Jacob and blessed him, thinking he was blessing Esau. Jacob received the blessing, regardless of who Isaac
thought he was speaking to when he said "Let people serve thee, and nations bow down to thee: be lord over thy brethren, and let thy mother's sons bow down to thee." Isaac was thinking "thee" meant Esau, and "thy brethren" and "thy mother's sons" included Jacob. It didn't matter...his hands were on Jacob, therefore those words applied to him.
I believe that the names "Lord Jesus", "Jesus Christ", "Jesus of Nazareth", and such titles refer specifically to Jesus as He is revealed in the Bible regardless of whether the person speaking the name has an accurate knowledge of Him. This includes Mormons, Jehovah's Witnesses, Unitarians, and others who hold to unorthodox views of Christ. "...that at the NAME of Jesus every knee shall bow..." (Phil. 2:10)
Keep in mind that there will be people at the judgment who have prophesied, cast out demons, and performed many miracles by using the name of Jesus...despite having never been known by Him. There is power in the NAME of Jesus.
Obviously we want to fully inform people of who Jesus Christ is and what He has done to give us salvation, but I would say this is done as the very first step in discipleship of new believers, rather than as an essential component of saving faith. The only fact about Jesus that I am certain a person needs to believe in order to be saved is that God raised Him from the dead; but seeing as how He lived 2000 years ago, anyone who calls out "Jesus please save me" must necessarily believe He is alive today, otherwise they wouldn't suppose He could hear them.
