Logically, the following two definitions are exactly the same:
For all $\epsilon >0$, there exist $\delta >0$ such that if $0<\vert x-a\vert<\delta$, then $\vert f(x)-L\vert<\epsilon$.
For all $\delta >0$, there exist $\epsilon >0$ such that if $0<\vert x-a\vert<\epsilon$, then $\vert f(x)-L\vert<\delta$.
But would people say that the second one follows a "bad notation", a "hard-to-read notation", "less-elegant notation", or an "unconventional notation"? I am trying to make the readers happy.
Yes, people would object. Although the choice of variable name doesn't matter mathematically, it can still be helpful or misleading. There is a general convention around the use of that particular pair of variables, and going against it will only confuse readers.
(Granted, every so often there is a good reason to go against such a convention ... but that tends to be pretty rare.)