'A' butterfly effect If a butterfly did not flap its wings some time ago, but instead decided to slide for that millisecond, can this cause a tornado on the other side of the earth if we just wait long enough? Does this perturbation die out, or average out in short time, or does it propagate and have grander and grander consequences?
(Clearly there are cases where it propagates, ie if its wings were artificially connected to trigger some nuclear bomb, but is it always so that it propagates?)
Is there some clear-cut criteria to determine whether a perturbation propagates or dies out?
Is the answer different if the world was purely classical versus QM?  
 A: If you take a well-behaved physical system and perturb it a little bit, then you expect the total behavior or your system to be changed only a little bit. You can quantify this by saying that if your initial perturbation is $\delta$, then the final perturbation can never exceed $\gamma \delta$ for some constant $\gamma$. In many cases, such perturbations manifest themselves as small oscillations around the unperturbed orbit. 
In a chaotic system, by definition, the deviation between the solutions for systems with minimally different initial conditions can become arbitrarily large. People first discovered this (at least a video on chaos theory I've seen in high school claimed so) when they tried to calculate the weather using some simple models. You quickly see that even very tiny changes in your initial conditions lead to completely different results for your forecast. 
Now, as was pointed out in a comment, a tsunami is not triggered by atmospheric disturbances, but by underwater earthquakes. Maybe you think of "tornado". Here the answer would be yes and no. On the one hand, the very fact that there is such a thing as "tornado season" (i.e. right now in the US) and certain regions on earth with heightened frequency of tornadoes (tornado alley in the US) shows that it is not absolutely arbitrary when and where these things occur. The grand scheme, as you could call it, will not be affected by a butterfly's actions on the other side of the globe. But if you set out to calculate then when and where up to the millimeter and the millisecond, you would have to take into account all the tiny atmospheric disturbances.
Here is a mathematical definition from the Wikipedia article on the Butterfly effect:

A dynamical system with evolution map ft displays sensitive dependence on initial conditions if points arbitrarily close together become separate with increasing t at an exponential rate. The definition is not topological, but essentially metrical.

A: The weather is a chaotic system and has the characteristic of having critical dependence on initial conditions. This in practice means that any forecast over 2 or 3 days starts to be flawed.
Chaotic systems are a feature of mathematics, and as such are present in classical physics:


*

*Magnetic pendulum

*Double pendulum

*Dripping
In practice, when people use the butterfly effect example, mean that the overall ignorance of the initial conditions determines the unpredictability of the larger system. In practice you should read that as: in order to be able to predict the weather long term, we would need to know a lot of detail, up to the level of butterflies flapping their wings.
It is not the case of a single butterfly determining the weather.
A: *

*The butterfly effect is to be taken metaphorically. The crux of it really is long-term unpredictability due to sensitivity to initial conditions. No serious meteorologist believes that the formation of a tornado could depend on whether a butterfly flaps its wings or not. Reliable short-term predictions of weather (over a few days) can be and are being made everyday without observing what butterflies do. But what the Butterfly Effect does say is that small changes in the initial data that you provided for your weather forecasting model will lead to wildly different predictions for the long-term (a week or more from now).

*The effect is purely classical and would still be observed in a completely classical world.

*Tsunamis are caused by earthquakes on the ocean bed, which can´t have anything to do with local weather. << I notice this is irrelevant after your question edit
A: You have to study a bit about deterministic chaos.. The illustration in the article is what might look like a butterfly, and the popular butterfly narrative may have its origin to the similarity, but there is a wealth of clear mathematical derivations that lead to statements like  the effect of " a butterfly waving its wings " .
It is an illustration that when one tries to solve the known dynamical equations of physical systems, very small changes in the initial variables condition can have enormous consequences to the solution output. The system of solutions can be studied mainly through the use of computers.
Thus your question does not have a meaning because you have taken seriously the butterfly waving its wings in Japan. It is the dynamical equations that might connect a small initial condition in Japan with a storm in London. No dynamical equations connect tsunamis with air disturbances.
