Is there a name for this class?
Your class $C_{1,\epsilon}$ is the same as RP. It is clear that everything in $C_{1,\epsilon}$ is in RP. In the other direction, take an RP algorithm for a problem, and amplify its success probability to be $1-\delta$. Now modify it to toss coins and reject with probability $\epsilon$ (or some constant $c &gt; \epsilon$) and only then run. If $\delta$ is small enough, then the overall success probability in the Yes case will be arbitrarily close to $1-\epsilon$.

The same argument shows that $C_{2,\epsilon}=coRP$, and so $C_{1\epsilon} \cap C_{2\epsilon} = ZPP$.