I've asked the same question (does anyone have written rules for two-foul pushout?) and received the same answers you have: lots of opinions about which is better; five-word "push out after any shot" answers; blah, blah. But no refences to written rules.
I guess back then no one wrote down rules. Must have been before the invention of written language.
The two-foul push out game will be lost forever after its current adherents die because no one will know the standard rules of the game. It's like wishing I had asked great-grandma more about the family history before she died. Once she was gone, it's lost forever.
I wish someone would write them down.
There were different ways to play. Some played "Two foul by the same player" others played "Any two" the way I liked to play by the way. In the south they often played "Honest attempt". If you didn't hit the ball they would make you shoot again but there was no BIH.
Then some places in the south they seemed to make the rules up as they went along especially if you were winning. Many black rooms had rules you never saw anywhere else. Same with 8 ball.
When you ran around in those days you just played by what ever rules they played by, you never argued. I often won even when they were making up rules trying to cheat me.