I would expect an algebra to be at least a ring, i.e. an algebra over a field, a vector space with a bilinear product. If you just say plain algebra, I think matrices and vectors or similar.

Most of the structures called algebras (e.g. boolean algebras) have at least two operations and these operations frequently interact via distributivity or something like that. Other examples are ring-like, like a sigma algebra in measure theory.

I've never really encountered anyone saying "algebra" for a generic algebraic structure. I think with this definition Cohn was trying to start a trend that didn't catch on.