Admittedly the experience of individuals with split brain is a bit out of my wheelhouse so I would love to be corrected on this. But it would seem to me that what you describe is not inconsistent with there being two separate consciousnesses present? If there are truly two consciousnesses present in a split brain, I would still expect that each one would describe itself as a single observer.
I think this is a little bit more complicated than that.
Split brain patients are fully aware of events where their executive function clashes. For instance, they can see how one limb operates out of sync with the other one. So how do they establish this awareness? Who is watching? Who is the one that verbalizes that fully conscious experience?
Certainly one could argue that this could be the result of an observer in the right or left brain verbalizing their subjective experience, but that creates several problems.
The biggest one is that for this to be the case there needs to be a commanding observer at any point in time and this would mean that one brain needs to communicate to the other when would it becomes the observer. That would require some unifying substrate that allows this cooperation to happen beyond the physical connection of both brains. This assumes a full separation of the left and right brain. There are also some theories (Gazzaniga experiment) that suggest that the left brain might be in this case the conscious operator simply because the left interpreter is capable of overwriting the right brain by default. But this experiment fails to explain the meta-observer experience that allows the subjects to effectively participate in the experiment and explain their experience with both right and left stimuli. The experiment correctly shows the existence of separate logic centers but can't explain why the subject is still experiencing his cognitive interpretation of reality as a single observer.
This is the fallacy of consciousness theorization. We don't have any good definition of consciousness because it can only be defined subjectively and uni-personally.
Empirically and logically we know that is implausible to have two simultaneously operating and fully aware consciousness. We know this because this duality doesn’t work at all with any rational interpretation of ourselves. We are one and only one. We are never two simultaneous people and there’s no thought or physical experiment that would allow us to test or imagine that.
Some people might have mental diseases that completely alter their consciousness and sense of identity but their experience is still a single observer experience.
Split brain patients are in fact the best example we have of consciousness being discrete and somewhat separable from brain function. This doesn’t necessarily mean that the brain isn’t the one producing the conscious experience but it certainly shows that the brain mechanisms that could produce consciousness are very different from the ones that enable other neurological functions.
The idea that a patient with two physically separated brains can have a relatively normal life (even when executive function clashes happen, one brain is still capable to overwrite the other) shows that there’s some unknown hypervisor function that enables their conscious experience.
Admittedly the experience of individuals with split brain is a bit out of my wheelhouse so I would love to be corrected on this. But it would seem to me that what you describe is not inconsistent with there being two separate consciousnesses present? If there are truly two consciousnesses present in a split brain, I would still expect that each one would describe itself as a single observer.