You should take a look at Theorem 3.4 (p 85) of Farb and Dennis' book Noncommutative Algebra. The statement is:
Let L/k be a finite extension of fields. Then K (x)_k L is semisimple for every field K >= k if and only if L/K is a separable extension.
That the tensor product is semisimple implies that its Jacobson radical vanishes. Conversely, any artinian ring with trivial Jacobson radical is semisimple. Therefore your equivalent formulation of separability is true.