Rigorous definition of efficient algorithm that $\epsilon$-refutes random 3CNF formulas
The words "exceptional" and "typical" have absolutely no formal meaning. Let's rephrase the definition using two other words, "Clinton" and "Bush".

We say that a (possibly randomized) algorithm $A$ $\epsilon$-refutes random 3CNF with ratio $\Delta$ if:


Given a 3CNF formula $\varphi$ with $n$ variables and $\Delta n$ clauses such that a $1-\epsilon$ fraction of the clauses (at least) is satisfiable, the probability that $A(\varphi)$ outputs "Clinton" is at least 75%.
With high probability, $A(\varphi)$ outputs "Bush" given a random 3CNF formula with $n$ variables and $\Delta n$ clauses.


What does the algorithm $A$ try to do? It tries to identify "Clinton" inputs, which are inputs that are at least $1-\epsilon$ satisfiable. At the same time, on a random input, which is with high probability not "Clinton", it should return "Bush". So it tries to separate the rare "Clinton" inputs from a random input.