He said that he chose to believe that the moon exists, even if he was not looking at it.
But he was wrong.
I think what your favorite theories are saying is that, if consciousness isn't observing something, it exists only in a state of probability. It only needs to be instantiated (with details filled in) once either observed by consciousness, or if the records of such observation are available to be viewed by consciousness.
We live in a virtual reality that appears to be lazy about creating details – that which does not need to be resolved at a fine level of detail, isn't. Perhaps its all to save CPU power? Who can say.
An unobserved tree in a forest isn't resolved into detailed existence until it is first observed. Then, if no record or memory is kept, the tree reverts back to an unresolved collection of probabilities – apparently to save space/time/etc. But if a record is kept of that tree's condition, and that record is available for consciousness to review, that pins the tree's actual configuration into place – aligned with the detail and reliability of that memory or recording.
As long as some consciousness is observing the moon – or a record of observation is being kept that is available for consciousness to review at a later date – the moon with all its details remains in place. Once there are no more records and no more consciousness, it drops back into a field of probability – a complicated enough definition to generate an appropriately-realistic moon quite similar to the one that used to exist should consciousness once again appear to observe the moon.
From an implementation viewpoint, it appears that everything that exists has a generator function that gets run to resolve the item into a unique, detailed object. The detailed object remains in place until it is no longer being observed, at which point it is deleted. If observation occurs again, the generator function is re-run and the object is recreated at whatever level of detail is visible to the observer.
That's as I understand it anyway. Its kind of how I'd do it if I wanted to make an infinite virtual reality, but I only wanted to expend resources that were strictly necessary for the consciousness to have its realistic experience.
Its pretty clear that reality exists to serve consciousness. Given that's the case, a related question is, what sort of effect can consciousness have on the generator function of each object? Can we, as conscious beings through intent, cause the generator functions to emit a level of detail that "bends reality" more in line with what we'd actually like to have happen?
If you know you have beers in the fridge, but you have quite forgotten exactly how many beers there are (and you have no receipts, etc), most likely, they exist in a state of probability only since they are out of view, and out of memory, and there's no need for the system to retain that level of detail, so it doesn't. Can you, through the exercise of your intent, create an extra beer or two, assuming that the total number falls into the range of probability available to that generator function?