How are these Floor Joists of the rooms either side of this stair case supported by it? ... Pic attached this time! ...?

The big beam in the basement was supported by a post in the basement. That post may be gone and now it is only the beam. You might want to add a big post. My husband just looked at your reference. He had never heard of ballon framing or knew anything about it, but it turns out it is the design that he used to build our (this) house. He came up with a way to raise the ceilings so that we could put ceiling fans in higher up. And he locked the floor and ceiling joists to the wall studs in such that they are stronger from hurricane grade winds. He says it is basically ballon framing, but he had never heard of it until now. So, he said to look at your connections and see if they are locked in the framing, more than nails.

Impossible to know without actually looking at the structure. The walls you’re referring to as load bearing may not be load bearing at all. The floor joists in question could run perpendicular to the joists at the top of the stairs. There’s absolutely no way of knowing from your picture.