You didn't include a link or date for the Dr. Phil material you quoted (that I can find anyway.) But I believe JP's first Dr. Phil appearance aired at the end of the 3rd week of Sept (around the 21st but it was taped earlier.) And the entire family made an appearance in early Oct.
If we assume what you quoted came from the earlier appearance, that's still 3 weeks after BL returned home. The point I was making in my post concerned what the Laundrie parents could have reasonably thought in very early Sept. And while JP spoke about the van as being "hers" on Dr. Phil, there was a great deal of speculation here on WS about who actually paid for the van, who paid for the insurance, and whose name and address was on the registration. Some occurred before the Dr. Phil appearance by JP but most was after. For example, see Threads #67, #65, #60, #48. Not everybody watched Dr. Phil and even after watching Dr. Phil, one could still have questions about who provided monetary contributions.
From pictures posted on BL's IG, it appears the van conversion happened at the Laundrie home. After watching that go on for months with BL doing much of the work, it would be perfectly understandable if the parents thought the van was jointly owned. It surely seemed to be a joint project.
Please note, I'm not saying the van was jointly owned. I'm saying it would have been plausible for the Laundrie parents to believe that it was early on. But the van was impounded on Sept 11, Brian went hiking on Sept 13, and GP's body was found was Sept 19. So by the time JP was on TV talking about ownership of the van, I'm pretty sure that issue wasn't a primary concern of the Laundries.
JMO