Well, since we can't (usually, reliably) return from an inactive state, and since we don't normally loose our cognitive state, there's no reason to differentiate between the two methods for determining self among humans. If we become inactive for long periods of time, we're dead, the question of who we are is moot.
This isn't true for the Marathon's AIs. Consider the question from, say Leela's perspective. Leela is defeated by the Compilers and deactivated, a state most of us would view as death. Yet, if I recall correctly, Durandal at one point tells us that he has brought her back on-line (and initiated a reset of her core logical systems, which I presume will undo what the S'pht did). Now, the entity that will thereafter exist will have Leela's memories, personality, sele-identity (i.e. will think she's Leela) - arguably, Leela will live again, even though a clear line of conintuity has been broken, she's been dead for a bit.
Now, where state as a definition for self fails, as has been pointed out, is that (given what we know) it is possible in theory for multiple AIs to have the same state, even though most people would consider them to be distinct individuals. Consider, what if Durandal hadn't deleted himself (or his original self) from the Marathon after installing himself on the Pfhor scout-ship. There's also the Tycho-clones, of course.
I can see Durandal believing that, as long there is an AI in state Durandal somewhere, he continues to live, even if there are breaks in continuity, interims where he's effectively dead. I also think Durandal might ascribe to this view out of necessity rather than conviction - ie, it allows him to be comfortable with being dead for short periods, a condition forced upon him anyway, so he might as well get comfortable with it.
