Ok, first of all, I'm sorry for my rude formulation, as a not native speaker it's sometimes hard to judge how something may sound while concentrating on correct grammar. I actually meant that from my viewpoint it's the same being. Anyway:
"it is shown and said in the real game that chara was the first human to fall in the underground" right, no comment on that fact
"DETERMINATION is the rarest soul type" I don't agree on that one. Why? Well, before things in Undertale even take place 7 humans are known to be dumb enough to climb a mountain without letting anyone know about it. Each one has also a unique trait. It is NOT KNOWN, how the state of proportional distribution of human soul traits on the surface is, nobody ever tells that. BTW, I just noticed that according to your opinion, if we extrapolate the known human souls, than DT is way more often present than any other trait.
Why I think about Chisk as the same person...
What we know for sure:
Toriel dragged Chara's body into the ruins.
No matter which path we choose the protagonist has at least one Chara's flashback at the point when Falling in the waterfall.
There is absolutely no evidence, that frisk actually fell when in the moment before the game started - no images, no text, not at least a simple sound like in watefall.
In Undertale's game files the red soul is called "spr_ourheart_0".
Remember how Toby Fox twittered that we should use our own name?
What we "know" about reincarnation:
It's the state of being reborn in a new body.
After being reincarnated one has some memories from the past life.
...which basically means that in year 201X we attempted suicide, failed, then succeeded, then much later we are reborn as "Frisk".
That's TL;DR: version of my interpretation and if you find anything that disproves that theory feel free to put it out here.