Are these two definitions equivalent?
A ring $R$ is called semiabelian by Yiqiang Zhou if its identity $1$ can be written as a finite sum $1 = e_1 + \cdots + e_n$ of mutually orthogonal idempotents $e_i$ such that each corner ring $e_iRe_i$ is abelian.
A ring $R$ is called semiabelian by Weixing Chen if every idempotent $e$ of $R$ is either left semicentral or right semicentral, where an idempotent $e$ of $R$ with complementary idempotent $f = 1-e$ is called left semicentral if $fRe = 0$ and right semicentral if $eRf = 0$.
Both definitions are proper generalizations of the notion of an abelian ring, and the first definition includes all semiperfect rings.
Follow-up: The answer from leslie townes shows that these definitions are not equivalent. More specifically, it shows that $(1) \not\Rightarrow (2)$. Does $(2) \Rightarrow (1)$?