Academy of Sciences of Heroesland

No, a three-dimensional sphere is our Universe. More precisely, I heard somewhere that the Universe is a three-dimensional sphere.

A four-dimensional one is a sphere that has volume but is described using four dimensions.
This, by the way, is not necessarily true. It is one of two versions derived from General Relativity. The second is that the Universe is an open, infinite space, described by three dimensions (that is, exactly how we see it with our ordinary perspective).

However, an ordinary sphere is a manifold of dimension 2, not 3.

Well, yes, but I don't like this terminology. It is obvious that closure cannot be ensured in two dimensions. Here, what is more important is how such a manifold looks to an observer, and not how it looks to objects inside itself, in my humble opinion. Although I don't argue with the dimension of the manifold; the terminology is accurate here, but with the dimension of the figure – well, yes))