Lightning strikes don't wipe people's personalities. Also during epileptic seizures people's neurons fire in abnormal ways, but then return to normal afterwards. So I'd say that the RAM isn't critical; humans can boot from disk.
We also probably have things that we forget/block/deny but are still there somewhere underneath, reachable only if we are put in the correct psychological conditions.
Like how deleting files doesn't properly wipe out the bits and with the right tools you can still recover deleted info.
A key concept in Netflix's cyberpunk series Altered Carbon is the 'stack', an advanced hard drive installed on the brainstem on which a person can save a copy of their consciousness. The main effect of stack technology is a form of immortality, because a stack can be installed into another body if the original body dies. But there’s another a major implication only hinted at during the first season of Altered Carbon: If you could choose your own body, would you go with the one you were born with? That's an especially important question for gender fluid or transgender people. The topic was only hinted at in the first season, but Altered Carbon creator Laeta Kalogridis told The Wrap it's something she would like to explore in greater detail.
There are some documentations of incarnation and related phenomenon, ranging from anecdotal to quite convincing. So I'd say that the disk might not be critical; humans maybe can boot from network.
Irreversible brain damage or death occurs when the brain is not getting enough oxygen.
If the brain continues its metabolism without oxygen it induces neuronal cell death.
The mechanism is called anoxic depolarization. Sodium–potassium pumps in the cell walls need to work to maintain K+ and Na+ ion gradient. When the pump runs out of energy it causes glutamate and aspartate concentrations inside the cell grow. When they grow past some critical threshold, it starts programmed cell death.
Cooling he body and brain can prevent this when the brain metabolism slows down to match the lower oxygen levels.
Electrocuting the brain like in electroshock therapy most likely wipes any electrical activity. That seems to indicate that the brain is able to recover after such an event.
The connections between the neurons is the architecture. The neurons themselves have state. If that state is wiped across the brain, I'd be surprised if the body connected to it could even still keep itself alive.
To the actual question though, neurons die/fail very quickly at their normal metabolic rate without an oxygen supply. You are certainly dead once that happens.
These procedures are all about providing some oxygen and dropping the metabolic rate to keep the neurons alive longer than they would unassisted.
What leads to brain damage, as I understand it, is lack of oxygen like all other organs and hypothermia reduces your brain activity and its consumption of oxygen but I always interpreted brain activity as an indicator of oxygen consumption and not as something that'd have a cause-effect relationship with brain damage.
Also, perhaps during such a cold state more cells are able to stay alive despite having less oxygen to work with? Maybe due to a reduction in how much energy they use or need per unit time? Just speculating.
I.e. is the state of your brain preserved just by the connections of the neurons? Or is the "RAM" also critical?