PSpace-completeness under PSpace reductions
Every language $X$ in PSPACE would be complete under your proposed definition, except for $\emptyset$ and $\Sigma^*$. You could reduce any PSPACE language&nbsp;$Y$ to&nbsp;$X$ by a reduction that first decides whether its input $w$ is in&nbsp;$Y$ and then maps to a fixed "yes" instance of&nbsp;$X$ if so, and a fixed "no" instance if not.

Note also that the actual definition of PSPACE-completeness requires that every language in the whole of PSPACE reduces to the complete language. The definition you've given is circular, since it defines PSPACE-completeness in terms of itself.