What prompts doctors to determine the time of a patient's death, as happened with the questioner's two relatives who died of cancer, even though only the Creator knows the unseen?

No one can definitively state the time of anyone's death, because this is from the knowledge of the unseen, which only Allah knows, as stated in His Almighty قول: "Indeed, Allah [alone] has knowledge of the Hour and sends down the rain and knows what is in the wombs. And no soul perceives what it will earn tomorrow, and no soul perceives in what land it will die. Indeed, Allah is Knowing and Acquainted." A doctor's informing of a person's impending death is merely a probability and a conjecture based on the symptoms of the illness and diagnostic tools; it does not provide certainty.