Well, that was fun. New phone has new colour space which requires updating Resolve which wouldn't install and required me to update MacOS. Recording and then viewing the first clip took a lot longer than anticipated.
Here's some initial observations as I find my bearings.
Resolve 20.2 only has Apple Log 2 in the Colour Space, and only has Apple Log in the Gamma.
If we assume that BM and Apply both know what they're doing then that means that Apple Log 2 gets a new colour space but keeps the same gamma curve, and iPhone 15 and 16 users get an upgrade to Apple Log 2.
In the default camera app you only get Log and HDR (whatever that is) in Prores mode. Not entirely sure what Prores flavour it is, but I'd guess HQ because the fine print in the settings to enable it says 1 minute of 4K 30p is 6GB which is 800Mbps (and would be 640Mbps for 24p) which is sightly lower than the 700 typically stated, but is definitely higher than the 471Mbps typically stated for 4K 24p Prores 422.
If you select Prores and Log, the image displayed is the log image.. no display conversion for you!
BM camera app gives tonnes more options. I'll have to investigate that further, but last time I looked it seemed to only offer manual shooting, rather than the auto-everything that I need when shooting fast.