Virmundi, thanks, you helped a lot! After finishing my PhD degree, I will actually have to work for my current institution in Turkey for 6 to 8 years. This is to pay what they spent for me. However, since it is a school where I don't plan to spend all my career, I will be looking for a better one, probably again in Turkey. Given that, yes, Oxford is definetely the bigger name having even a magical name here. It is an object of jokes regarding education, like "did not we study there while Oxford was here?"
And yes, the UC Davis Professor is a great fit as an advisor. But the Oxford Professor is also a popular name among the historians of the Middle East. He is an American with a Harvard PhD, not-great-but-ok fit. The reason I am leaning into Davis is I am just sure about helpfullness and personality of the UC Davis Prof. as I have met him before. I care advisor-advisee relationship too much since I have a bad experience of it at the school where I am finishing my Master's. In this regard, I think that I should try to develop a further communication with the Oxford Professor too.
As a result, I will apply to both and decide after results were announced.
Thanks for your comment.