Once upon a time this wasn't so. You dropped, you died. Done. With the advent of invasive monitoring and CPR however the definition of death has become much more variable, leading predictably to lots of problem including in halacha.
The problem with using the Talmud to determine the exact definition of death is that the cases discussed by the Sages all occured in the absence of an ability to monitor or accurate assess the internal workings of the bodies being examined. The Sages had no way of measuring blood pressure, never mind brain waves or the electrical activity of the heart. Thus the classic case that is always reference, from Yoma, discusses the cessation of respiration as the criteria for death. How the Sages would have paskened today with the more advanced understanding of anatomy and physiology is what leads to the divergent opinions between the poskim. Some still see cessation of respiration as the gold standard and now that we understand that respiration is controlled by the brain stem this means that brainstem death is the criteria for true death. Others hold that the Sages were looking at circulation and therefore cardiac activity is the gold standard. And of course, this being a halahic dispute there is much rancor and bitternes on both sides. After all, if the respiration definition of death is correct then the circulation folks are indirectly killing patients who require transplants and if the cardiac definition is correct then the people relying on brainsteam death are murdering patients for their organs.
What is not helpful is when terms get mixed up. It is important to remember that "brain death" and "brain steam death" are very separate things. According to both the respiratory and circulatory positions, destruction of the cerebral cortex with a residual functioning brain stem is not a true death. This is important to remember when the subject comes up and often gets blurred with the term "brain death" acting as an inappropriate catch all. This is also important because in the secular world there are those for whom properly defined "brain death" is a form of death leading to a person becoming an organ donor.
Thus Rav Natan Slifkin's critique of a recent Wall Street Journal article both emphasizes but also aggravates the discussion.
In my opinion the comparison with the lizard tail is what is irrelevant. A severed tail is only part of an organism, non-essential to the life of the creature. The brain is not like that at all and whether it is considered physiologically severed is of crucial importance. I'm surprised that Rav Slifkin did not bring the mishnah in Oholos which specifically discussed decaptitation as a form of death despite signs of life from the body. That is what Rav Moshe Feinstein, zt"l, used as his definition in a later teshuvah and would have fit more in the discussion of brain and brain stem death.
The article writes about how brain-dead people have "more in common biologically with a living person than with a person whose heart has stopped. Your vital organs will function, you'll maintain your body temperature, and your wounds will continue to heal. You can still get bedsores, have heart attacks and get fever from infections." It talks about how they "react to the scalpel like inadequately anesthetized live patients, exhibiting high blood pressure and sometimes soaring heart rates."
This is all entirely true. It is also entirely irrelevant.
Physiological processes do not always denote life, and reactions are not the same as feelings. The detached tail of a gecko can move around with complicated motion and respond to an external stimulus, but clearly the gecko does not feel anything. Even a properly anesthetized patient can respond to the surgeon's scalpel and have their blood pressure go up, but that does not mean that they are feeling anything.
Another point to recognize is that the whole pain argument is out of place. Pain is a conscious reaction to a physical insult like a scalpel cutting into the body. In the absence of pain a person can still have an automatic response to such an insult resulting in measurable physiological responses but that doesn't mean he feels the pain, especially if his higher cerebral centres have been destroyed. Yes his heart rate will go up but there is no consciousness that feels the pain.
The Wall Street Journal article therefore does not prove its point but both it and the rejoinder seem to miss a bit by not clearly describing the difference between brain death and brain steam death.