Symmetric property of a binary relation
Solution 1:
No. The definition is what you wrote, and then you deduced from it that$$\left(\forall (x,y)\in E^2\right):x\mathrel{\mathcal R}y\iff y\mathrel{\mathcal R}x.\tag1$$But that doesn't make the definition wrong somehow. It only means that if we had decided to adopt $(1)$ has the definition, nothing would change.