Let me say everything I have to say about this theory of yours.
First: Just because it says stuff like 'their name' doesn't mean that it's Chara. From what I can tell, Toby Fox used 'they' as a build-up. (If it was just 'him' and 'his name' it would be too obvious, and the scene wouldn't be as powerful as it is right now.) And may just be what I think, but let's turn your logic back on you: Napstablook is also called 'they'. Therefore, the entire point is to save Napstablook. See? It doesn't make sense.
Second: Why the heck would you want to save Chara? Let's see, you can't access your SAVE file and you're up against a god trying to figure out how to break the barrier and set everyone free without hurting anybody. What do you do? You save the person who's trying to KILL EVERYBODY AND DESTROY THE UNIVERSE? I don't think so.
Third: Even if it IS Chara, that just doesn't make sense for the rest of the encounter. You have a nice scene, and it really tore my heart to bits, but the scene has no Chara in it. It is all saving Asriel. And the 'someone else' is replaced by Asriel. Why would it be Chara?
Fourth: You can argue that the slideshow is mostly about Chara, which is true. But there's a reason for that. It shows Chara in what looks like pain, and then it shows Asriel finding them, helping them up, taking them to his house, and playing with them. I don't think anything could Asriel of who he once was more than the friend he used to be to Chara, and the scene shows it. You don't achieve the same results without Chara in there.
Fifth: Chara isn't even showed or mentioned at all (unless you count 'their name', which I don't) in the True Pacifist ending. If you really did need to save Chara, there would be at least a tiny mention.
And yes, the whole narrachara theory has nothing to do with this. So please keep it out.