The difference lies in what can bring you back to life, and in this case how much exhaustion you have afterwards.
Oh, I see. Yes, this is an interesting discussion. I'll address this in a bit, but first I'd like to respond to the rest of your post:
"Death" can mean completely dead, or it could mean dying.
No, "dying" is what happens when you are reduced to 0 hit points:
If you're "dying," you're in trouble. If it gets bad enough (three failed saves), you die.
Exhaustion is essentially just a slower version of "dying":
If you're exhausted, you're in trouble. If it gets bad enough (six levels), you die.
These are two separate systems that do a very similar thing. Maybe it would be more elegant if they were connected in some way, but they kinda added the exhaustion rules at the last minute, so it's a little sloppy. D&D 5e is unfortunately not seamless.
It seems more than a little bit silly to me to tell my players "So, yeah... if you're beaten unconscious with an axe your buddy there can stabilize you, but if you lose consciousness in 'frigid water' you better hope your chilled corpse gets dragged back to the temple district with full pockets."
Losing consciousness in extreme cold is like losing consciousness in a room full of goblins that repeatedly stab you. In the cold, it's the sixth level of exhaustion that kills you. In the goblin room, it's the third failed death save that kills you. Either way, you're dead if you're there too long.
And while I'm at it:
If you're dead by exhaustion, then there's no point where you're "almost dead by exhaustion" which the spell could set you back to and which the spell could prevent.
Yes there is. Level 5 Exhaustion is "almost dead by exhaustion."
The difference lies in what can bring you back to life, and in this case how much exhaustion you have afterwards.
Spells like Death Ward and Raise Dead were clearly not written with the exhaustion rules in mind (probably because, as stated above, the exhaustion rules were added after the fact). So there is a hole in the rules that we need to patch. I like the Death Ward solution floated upthread, where Death Ward can prevent you from gaining the final level of exhaustion.
As for Raise Dead--it brings you back at 1 hp, completely reversing the "dying" system, so I'm tempted to say it brings you back at 0 exhaustion. But the last paragraph says "coming back from the dead is an ordeal," and lays out an exceptional system that seems very similar to exhaustion. Looks like the designers forgot to revise this after they came up with the exhaustion rules. My solution: Remove the weird -4 penalty thing. The target comes back at Level 5 Exhaustion (regardless of their exhaustion level before death). Rest easy son, you've had a busy day.
Resurrection has a similar clause, so I'd use the same solution. True Resurrection brings you back good as new, so I'd say you come back with no exhaustion.