There are several ways to run a given elementary cellular automaton in a stochastic way:

by giving for each of the eight local configurations 000,100,010 and so on a probability by which the rule is applied

by applying the rule for all local configurations with the same probability

by applying the rule not in parallel for all cells but sequentially, picking cells randomly one after the other, then updating them deterministically.

I tried out 2. and 3. for finite cellular automata on $\mathbb{Z}/k$ and came to surprising results. The class III rules 18, 22, 122, 126, 146 have quite similar space-time diagrams and are classified by almost all classifications into the same class. (I used this online tool to create the diagrams.)

Applying stochastic mode 2 with probability 0.9 yields

Applying stochastic mode 2 with probability 0.5 yields

Applying stochastic mode 3 and creating a snapshot after every $k$-th step (when $k$ cells have been updated, just like in parallel mode) yields

The effects are similar but different nevertheless. For example, for rule 126 in stochastic mode 3 there are (almost) never white regions wider than one pixel. What's more striking is the quite different behaviour of the five otherwise so similar rules, which becomes apparent even stronger in stochastic mode 3.

Has anyone an idea how to explain (or possibly predict) this?

For the sake of completeness here stochastic mode 2 with probabilities 0.1 and 0.0:

non-zeroupdate probability tendtowardszero (while scaling time so that the average number of updates per cell per scaled time unit stays constant) is obviouslynotthe trivial process shown in your last image, where the update probability is identically zero and thus nothing ever changes. $\endgroup$