The Lovász Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an assignment of the variables that avoids all bad events if the LLL criterion $ep(d+1)<1$ is satisfied. In this paper, we study the dependency of the distributed complexity of the LLL problem on the chosen LLL criterion. We show that for the fundamental case of each random variable of the considered LLL instance being associated with an edge of the input graph, that is, each random variable influences at most two events, a sharp threshold phenomenon occurs at $p = 2^{-d}$: we provide a simple deterministic (!) algorithm that matches a known $Ω(\log^* n)$ lower bound in bounded degree graphs, if $p < 2^{-d}$, whereas for $p \geq 2^{-d}$, a known $Ω(\log \log n)$ randomized and a known $Ω(\log n)$ deterministic lower bounds hold. In many applications variables affect more than two events; our main contribution is to extend our algorithm to the case where random variables influence at most three different bad events. We show that, surprisingly, the sharp threshold occurs at the exact same spot, providing evidence for our conjecture that this phenomenon always occurs at $p = 2^{-d}$, independent of the number $r$ of events that are affected by a variable. Almost all steps of the proof framework we provide for the case $r=3$ extend directly to the case of arbitrary $r$; consequently, our approach serves as a step towards characterizing the complexity of the LLL under different exponential criteria.