Doesn't this mean that at least one of the results is wrong?

No, I think one of the fundamental insights of stochastic calculus is that the addition of noise to a process changes the trajectory in a non-trivial way.

In finance, for instance, it leads to the concept of a "volatility tax." Naively, you might think that adding noise to the process shouldn't change the expected return, it would just add some noise to the overall return. But in fact adding volatility to the process has the effect of reducing the expected return compared to what you would have in the absence of volatility. (This is one of the applications of the result that the original article talks about in the Geometric Brownian Motion section.)

Just to add to this, the reason that the things are different is, stochastics as a subject is trying to do calculus in the presence of noise, and what noise does is, it makes your function nondifferentiable. You would think that you cannot do calculus, without smooth curves! But you can, but we have to modify the chain rule and define exactly what we mean by integration etc.

So the idea is “smooth curves do X, but non-smooth noisy curves do Υ(χ) where χ in some sense is the noise input into the system, and they aren't contradictory because Y(0) = X. (At least usually... I think chaos theory has some counterexamples where like the time t that you can predict a system’s results for, is, in the presence of exactly 0 noise, t=∞, but in the limit of nonzero noise going to zero, it's some finite t=T.)

Kinda. The differential operator in quantum Ito calculus can be applied to mathematical objects that the normal differentials aren’t properly defined on, such as stochastic variables.