I read that post once again, and I think I found the fault.

The physical interface is, of course, the bloody spike that's plugged into the victim's head. The Matrix personality is not a "construct". You seem to be thinking of it as some kind of avatar, but it isn't. Once you are plugged in, this is you - your mind and personality as you've perceived yourself from pod to man.

The reason people die in reality when they die in the Matrix is that the mind perceives a physical injury, is broadcast awful pain through the spike, and is told by every nerve that it's dead. So it complies. This is part of the Matrix psychology. People who realize that the spoon is not there may recover, because their mind knows that it's being fed the wrong information.

So what does Smith do? He infects other entities in the matrix. Programs is easy, but he seems to be able to infect humans, too. When he sticks his hands into you, it translates in reality to a feed of data that goes through your feed spike, and tells your brain that you are now Smith. In effect, he reprograms your brain. Your synaptic connections get severed and re-arranged. It's a powerful re-programming, but that's what he does. Then the person is disconnected from the Matrix, but his brain remains re-arranged.