Did not the verses on repentance, baptism and continued faithfulness not apply to the jailer?
I doubt Peter taught the jailer OSAS, FART or the "sinners prayer".
Peter wasn't even there. It was Paul and Silas. Here is the account:
(Act 16:25) But at midnight Paul and Silas were praying and singing hymns to God, and the prisoners were listening to them.
(Act 16:26) Suddenly there was a great earthquake, so that the foundations of the prison were shaken; and immediately all the doors were opened and everyone's chains were loosed.
(Act 16:27) And the keeper of the prison, awaking from sleep and seeing the prison doors open, supposing the prisoners had fled, drew his sword and was about to kill himself.
(Act 16:28) But Paul called with a loud voice, saying, "Do yourself no harm, for we are all here."
(Act 16:29) Then he called for a light, ran in, and fell down trembling before Paul and Silas.
(Act 16:30) And he brought them out and said, "Sirs, what must I do to be saved?"
(Act 16:31) So they said, "Believe on the Lord Jesus Christ, and you will be saved, you and your household."
(Act 16:32) Then they spoke the word of the Lord to him and to all who were in his house.
(Act 16:33) And he took them the same hour of the night and washed their stripes. And immediately he and all his family were baptized.
Verse 32 says that they spoke the word of the Lord to him. Whether Paul and Silas taught the jailer OSAS, FART or the "sinners prayer" depends on whether such things are in the word of God. (Incidentally, I have no idea what "FART" is in this context. I'm fairly certain you aren't using the slightly rude term for breaking wind

).
Also we see that the jailer and his family were baptised
after believing:
(Act 16:33) And he took them the same hour of the night and washed their stripes. And immediately he and all his family were baptized.
(Act 16:34) Now when he had brought them into his house, he set food before them; and he rejoiced, having believed in God with all his household.
That ties in with what Philip told the Ethiopian who requested baptism:
(Act 8:36) Now as they went down the road, they came to some water. And the eunuch said, "See, here is water. What hinders me from being baptized?"
(Act 8:37) Then Philip said, "
If you believe with all your heart, you may." And he answered and said, "I believe that Jesus Christ is the Son of God."