My culture is definitely Patsy's and not some random strain. You can definitely taste it in my version. I just bought it, fed it and it kept on growing. I think they use old dough. But I'm not 100% sure. I doubt they keep a separate starter. I'm also pretty sure that they add a baker's yeast booster, which I do too. I talked to the long time waiter and he wouldn't get into too much detail, but this is what I surmised. As far as dough vs battery starter, I never thought that it made too much difference either way. I read the book they sell at sourdo.com and he talks about some difference, but I never played around too much in that direction.
I had another culture that I had for a little while which came from a pizza place across the street from Johnny's in Mt. Vernon. It was obviously the same culture as Johnny's . Yeast cultures can dominate a neighborhood. You could just taste it was the same. Johnny's is super secretive and won't give me anything, but I happened to pop into a place nearby and I could tell, even though the pizza was not great, that the basic underlying taste of the culture was there. Unfortunately, that culture died after a few weeks. I opened it up and it was patsy's culture. It had been taken over. I might try again someday. It was very good.
I also bought 2 cultures from sourdo.com, but these died too. I could probably do much better now, because I'm much more experienced than I was then.
I also once baked a dough from Patsy's in my oven here in Atlanta. I took it straight from the plane to the oven. That dough was unbelievable. It windowpaned, and blistered 10 times more than any other dough I've ever seen. You could spread it into phyllo dough and it would have unpoped bubbles in it. It was by far the most elasatic dough. But when I baked it, it wasn't fabulous. I was surprised. The taste lost something in my oven. I know I don't have a coal oven, but I had become convinced that it was more about the heat distribution than anything. Now I realize that the burning coal does add something I can't reproduce. However, my current pies have overcome some of that. I age mine more than Patsy's and this helps. I'm pretty damn close right now, especially recently since switching to the DLX mixer. Huge improvement.