I would only pass the necessary information. I tend to use Resource for stripping/passing only necessary model information, and limiting database calls to only return the information needed.
I do not know how livewire works in this senario, but blade (in my understanding) only passes the information actually used. Inertia doesn't discriminate in the info it passes.
Maybe showing some age, but I do not like passing any information that is not actually needed. Back when we still had a majority on dial up and slow bandwidth services, payloads needed to be as slim as possible. I've kept that throughout the last 25 years and personally find it very bad practice to send unneeded information like full models if I only need an id or name property when possible.
Yes, if you are correct, docs should be updated!