I was reading about L1-norm where I came across the term Cross-Polytype. In the link for Cross-Polytope, examples are given for Cross-Polytope in different dimensions. What I understood from the examples is that the sides should be formed from equal length line segments. In that case, can a cube be called a Cross-Polytope in 3 dimensions? and in that case can any shape with equal length sides be called a Cross-Polytope? for instance, a square pyramid.
It's made clear in the Wikipedia article you link that the cross-polytope in $3$ dimensions is the cube:
Also from the article, the definition makes clear that there is only one cross-polytope in $n$ dimensions, the object whose
Since that doesn't describe a cube, the answer to the question in your title is simply no, and similarly to your other questions: there is only one $n$-cross-polytope for any given $n$.