If we don't know that the true equation is x(t+1) = f(x(t),u(t)) but have to infer it from data, then observing that such an equation seems to hold true for some particular u and x does not mean that u causes changes to x.

It may simply be a predictor of x, as in the 'barometer readings may precede and imply a storm but not cause one' example given by others above.

Differential equations say nothing about causality they just describe what happens. Please remember that Pearl literally wrote the book on this topic before assuming he missed an elementary counter example.

I feel like reversiblity is a big complication here. Very interesting and subtle area.

I don't see the relevance. You can have (non-differential )algebraic equations that involve a "time" parameter. They can be non-reversible in the sense that I think you mean - that knowing a point "back" in time tells you values in the future but knowing a value in the future does not (uniquely) determine the solution before that point. [For example x^2 = t^2 for t <= 0 and x^2 = 0 for t >= 0. There are two continuous solutions x(t) which are equal at say t=1.]

For differential or non-differential equations, they're still just describing how something is and not the causality behind it. It's always possible that the equation is merely a result of hidden variables and there is no casual relationship between any two points of the solution in any meaningful sense.

I meant in how people talk about "causality", it is always poorly defined. It seems like there is a implicit assumption about irreversible systems at play.

Did you mean that, under reversibility + determinism, the future “causes” the past just as much as the past “causes” the future?

For some closed physical systems this is pretty much true, but I think the way people think about the world is at odds with this. That is why we find things like this so surprising https://youtu.be/p08_KlTKP50

It gets even murkier when you think of statements like "Hitler coming into power caused world war 2". There are so many things going on in that system that it couldn't possibly be true (e.g. if you change Hitler out for another person maybe world war 2 still happens), but works as a plausible line of causal reasoning for a lot of people.