You got it precisely correct.
There are surely Saudi’s that can object to permanent American military presence.
It can be assumed that the king/Crown Prince requested the removal because of intelligence indicating the threat no longer existed
It does not reasonably follow the Patriots are being removed to a higher threat location
I thought it was telling that the missiles protecting the refinery were removed. The ones protecting the air base remain. My guess is that the forward staging will remain as a matter of convenience.