The butterfly effect is formally captured mathematically. Consider a chaotic system (such as a mathematical equation of the weather) and an initial state. If we use those equations, we can calculate what the state will be at some future time (say, 1 month in the future). For this deterministic system, that is the only possible state we can be at in 1 month.
Now perturb the initial state slightly. Pick a "nearby" state, such as a state where the butterfly is about to flap its wings instead of staying still. Now use the same equations to calculate the state 1 month in the future. For this deterministic system, there will be one and only one possible state that we can be at in 1 month. However, because the system is chaotic, this state may be drastically different from the start in the first scenario.
Now this mathematical result points out two interesting physical outcomes. The first is to note that the state of the weather in 1 month is incredibly hard to predict. In theory, it's future state depends on how polarized or depolarized a few neurons within a butterfly's brain are -- the neurons which control wing flapping. The idea that the future state of a system as large as the globe is highly sensitive to such a small detail is I, in my opinion, a pretty significant philosophical statement.
The other major result of this has us noting that we never measure anything perfectly. There's always some error in our measurements. This means we cannot predict the future state of such chaotic systems accurately. It is a practical limitation to the capabilities of science.
It also implies that if we got the laws of the universe slightly wrong, we could measure everything perfectly and still not be able to predict the future state of chaotic things like the weather. Or, as you suggest, if the consistency of the world's laws were suspended for a brief moment in a tiny place, the effects could still change very large systems.
As a final thought, the butterfly theory does indicate that if you were able to suspend the rules of physics in a deterministic chaotic universe and change one thing to a state that it wasn't, you would indeed cause widespread changes in the overall state of the universe. God "changing a beetle" may indeed result in a tornado. One might say "the beetle caused the tornado," or maybe one might just say "God caused the tornado, and used a beetle to do it."
As for cause and effect, I think mathematicians generally agree that it is an ineffective concept in chaotic systems. It's not impossible to use, as you noted with the beetle causing a tornado, but it becomes less meaningful. I find the more meaningful use of cause and effect is in situations which are stable in the presence of perturbations, where identifying a "cause" permits you to mostly ignore large numbers of other details which won't have a large change on the effect. In chaotic systems, you can't ignore the other effects (because they could cause huge changes as well), so the benefit of identifying a "cause" is limited.