Multicore Doorbell Current Core/Other Core Mislabelled
In both the SDK, and the SDK reference, the multicore doorbell check functions have been incorrectly documented as checking the opposite core than it should be checking.
https://github.com/raspberrypi/pico-sdk/blob/a1438dff1d38bd9c65dbd693f0e5db4b9ae91779/src/rp2_common/pico_multicore/include/pico/multicore.h#L380-L396
By "SDK reference" do you mean https://datasheets.raspberrypi.com/pico/raspberry-pi-pico-c-sdk.pdf ? Note that the function documentation in that PDF is automatically generated from the Doxygen comments (that you've mentioned above) in the SDK itself, so when the SDK gets fixed the PDF will get fixed too 🙂
Please feel free to submit an appropriate PR against the develop branch.