There is a difference though, and those examples aren't equivalent because whats ultimately feared hasn't changed
A fear of dying alone the core fear isn't dying itself that's the fear, or doesn't rely on dying itself as the fear, it's that happening while their all alone
While the fear of death comes from the concept of death itself
Dying alone in this case is more akin to disappearing with no one she truly loves or cares for at her side, if she had someone she loves nearby then the fear of being alone would be lessened and the fear of death would remain irrelevant
Maybe it sounds like I'm repeating myself, but really that is just a fear of death. Like, the examples I've given
are "equivalent because whats ultimately feared hasn't changed." That's the point I'm trying to make. In a "fear of dying alone" the ultimate fear is
death, not being alone. Being alone is just the condition that, when removed, will alleviate your fear.
Like with the parachute example, the ultimate fear is "falling off a cliff," not "not having a parachute." This is obvious, of course, but its the same idea as with the "fear of dying alone." There's no fundamental difference between the structure of these fears.
Like, you say that "the fear of death comes from the concept of death itself" but that's like saying "the fear of falling comes from the falling itself," as if being afraid of splattering against the ground means that they're not actually afraid of falling. But if someone's standing in front of a cliff and you're refusing to give them a parachute, then they're still afraid of falling! Even if they would stop being afraid should you give them the parachute.
So saying "the fear of death comes from the concept of death itself" is nonsensical unless you're going to posit that every other fear works in the same way, which no one would reasonably believe. So what makes death special? Why is death the only fear that is only valid unless you're afraid of death in a vacuum without context? Why is the fear of death the only fear that has to come from the raw concept of death itself?
Personally, I think that this is a result of being afraid of being afraid of death. (thanatophobophobia?) Death is inevitable and unstoppable, so being afraid of it means being accepting a fear that will never leave you. And people don't want to accept that, so they look the other way and pretend they aren't afraid.
But anyways, yeah, if you're afraid of "dying alone," then you
are afraid of "dying." The fact that removing the latter half and having someone by your side when you die will result in you not being as afraid doesn't mean that you weren't afraid of death in the first place.
And I will acknowledge that I haven't really said anything different than in my previous comment, but in the first place I don't really think that you said anything in your comment that wasn't already addressed by my previous comment.