Plot. Nothing but that.
Movie script writers try to make a challenging story, not a realistic one. It suffices if the fridge logic holds until the movie is over.
If you could get shot in the matrix and just reload, how would the movie be thrilling? If you could make backup copies of your brain and load ten thousand Morpheuses into the matrix (similar to the "guns, lots of guns" that they just create out of nowhere), what would you stop from taking over the world?
Why bother with a haphazard rescue operation in a military-held building if one of the protagonists is held captive by agents? Just pull the plug, and reload.
The whole movie wouldn't make sense that way.
There is no physiological reason why you should die if you get sensory input such as "got hit by a bullet" when there is no actual wound. Yes, an elderly person with considerably old coronary arteries and a heart history might accidentially die from the shock, but that is a different story (and not even old people with a heart disease are that reliable at dropping dead).
Although respiration is certainly dependent on a semi-autonomous region in the brain, it doesn't work that way. On the contrary, the bodies of real dying people are gasping to the last moment as the respiratory center tries to keep oxygen supply up at all cost. Yes, you can willingly hold your breath, up to some limit. That's the "semi" part in semi-autonomous. You can't hold your breath until you die, the "hard wiring" in your respiratory center won't let you.
Similarly, the brain does have some influence on the heart rate, though indirectly (via the sympathic and vagal nerves). Stopping the heart by one's will or such, however, is simply impossible. It's a tale from the movies, and it's a cool plot for James Bond to escape from the secured ICU in Die Another Day, but that's all it is. Your heart will not just stop because you wish that to happen or because you think you're dead.
Also, a non-damaged heart that has been stopped (say, by an electric current during the vulnerable phase) can trivially be reset using defibrillation and will autonomically start beating again. Unless there is physical damage to the conduction system (or to the heart in general), or unless you inject potassium chloride or another deadly poison, there is no way you could prevent a healthy heart from beating. It's what the heart does until you die, literally.
From an information technology point of view, the idea that your mind is "gone" when the cable is cut is wrong, too. This is an often-used movie trope. You make a copy of some data and you have the only existing copy on that USB stick in your pocket. To save the world, the good guys have to retrieve this exact copy. Thrilling, but not the way things work. Making a copy or accessing information doesn't "pull it out".
In order for things to work the way as it's depicted in most movies, you would have to copy data, and then explicitly erase it from the original store. Dependent on where you copied the data from, you may not have permission to do so, or it may simply be physically impossible (e.g. read-only medium).
The human brain in particular comes very close to such a read-only medium. It is not designed for being erased on demand (though one could argue that short-term memory may be considered "eraseable", in some twisted interpretation, by hitting a person hard enough with a hammer).
While nobody (me included) knows or understands exactly what the human mind (or memory) is, it is a very complex, redundant and error-corrected, yet selective and lossy permanent store (lossy both in respect of applying a level of detail on information and in respect of actually losing memories over time). Memory is stored after going through a complicated filter machinery which is not fully understood and is linked with other related and unrelated sensory information.
Memories and abilities can be temporarily suppressed, but cannot be erased other than by physically damaging the brain, although they can be forgotten over time (but forgotten information can be restored from singular fragments, or with very little practice). In one word, it doesn't work like a computer's memory at all. Not in any way.
Now, whether or not the Matrix actually copies the minds of the humans or whether the brains are merely used as "processors", the information in the brain is not affected by that (other than new memories being added, obviously). When someone pulls the cable, your mind is not suddenly "gone", this just isn't how things work.
Your mind doesn't split up and half of it is trapped in a different dimension (Star Trek TNG, "We'll always have Paris"), and you cannot simply be "reprogrammed" to be someone completely different with different memories at the press of a few buttons ("Total Recall"). It's a cool idea for a compelling movie, but that's it.