We consider recent formulations of the algorithmic Lovasz Local Lemma by Achlioptas-Iliopoulos-Kolmogorov [2] and by Achlioptas-Iliopoulos-Sinclair [3]. These papers analyze a random walk algorithm for finding objects that avoid undesired "bad events" (or "flaws"), and prove that under certain conditions the algorithm is guaranteed to find a "flawless" object quickly. We show that conditions proposed in these papers are incomparable, and introduce a new family of conditions that includes those in [2, 3] as special cases. We also consider another condition that appeared in [3] in the context of sparse k-SAT formulas. This condition imposes a constraint for each variable of the problem, whereas traditional LLL formulations impose a constraint for each clause. Achlioptas et al. handled the variable-based condition via a reduction to a different condition and then applying a single-clause backtracking algorithm. We propose a new condition that directly captures the sparse k-SAT application considered in [3], and allows the use of the standard local search algorithm (which offers important advantages such as parallelization). Finally, we extend our previous notion of "commutativity" from [20] and prove several implications of commutativity using some new tools that we develop. In particular, we simplify the result of Iliopoulos [16] about approximating the LLL distribution.