The notion of PAC in approximation algorithms
Approximation algorithms don't make sense here, because here the output of the algorithm is 0 or 1.  (0 represents "the input is not in $L$", 1 represents "the input is in $L$".)  Approximation algorithms are useful when the output is a continuous variable, as then we can ask for the algorithm to output something that is close to the correct answer, but the concept is not useful when talking about algorithms that output a boolean.

Also, in your proposed definition, once you have fixed an algorithm $A$ and a distribution $D$ over $L$, the condition $ERR(A,D) \le \epsilon$ is either true or false.  It's not an event that you can assign a probability to.  So, it doesn't make sense to ask it to hold with probability at least $1-\delta$.